Seb

Toota Tarik

Keywords: entiers sans facteur carré, équirépartition en progression arithmétique, sommes d squarefree integers, arithmetic progressions, short exponential sums

Cette thèse concerne quelques problèmes liés à la répartition des entiers sans facteur carré dans les progressions arithmétiques. Ces problèmes s'expriment en termes de majorations du terme d'erreur associé à cette répartition.

Les premier, deuxième et quatrième chapitres sont concentrés sur l'étude statistique des termes d'erreur quand on fait varier la progression arithmétique modulo q. En particulier on obtient une formule asymptotique pour la variance et des majorations non triviales pour les moments d'ordre supérieur. On fait appel à plusieurs techniques de théorie analytique des nombres comme les méthodes de crible et les sommes d'exponentielles, notamment une majoration récente pour les sommes d'exponentielles courtes due à Bourgain dans le deuxième chapitre.

Dans le troisième chapitre on s'intéresse à estimer le terme d'erreur pour une progression fixée. On améliore un résultat de Hooley de 1975 dans deux directions différentes. On utilise ici des majorations récentes de sommes d'exponentielles courtes de Bourgain-Garaev et de sommes d'exponentielles tordues par la fonction de Möbius dues à Bourgain et Fouvry-Kowalski-Michel.

Mots-clefs :

Table des matières

Table des matières

Introduction I Répartition de fonctions arithmétiques

En théorie analytique des nombres, la répartition de suites arithmétiques en progressions arithmétiques joue un rôle central .On considère A un ensemble infini d'entiers strictement positifs. Pour X ≥ 1, pour a, q des entiers tels que (a, q) = 1 et 1 ≤ q ≤ X, on définit S A (X, q, a) := #{n ∈ A, n ≤ X, n ≡ a (mod q)}, et S A (X, q) := #{n ∈ A, n ≤ X, (n, q) = 1}.

Plus généralement, pour une fonction f : Z >0 → R ≥0 , on définit les deux fonctions de comptage S f (X, q, a) := n≤X n≡a (mod q)

f (n) et S f (X, q) := n≤X (n,q)=1 f (n). (1)
On note que l'on peut récupérer la notion précédente en faisant f = 1 A , où 1 A est la fonction caractéristique de l'ensemble A ⊂ Z >0 . Une première question qui se pose est celle de trouver un équivalent asymptotique pour S f (X, q, a). Si la fonction f est suffisamment régulière, on peut s'attendre à ce que pour tout entier q ≥ 1 et tout a tel que (a, q) = 1, on ait S f (X, q, a) ∼ 1 ϕ(q) S f (X, q), (X → ∞).

Dans ce contexte, on appellera S f (X, q) le terme principal. Par exemple, quand

f (n) = 1
n , si n est un nombre premier, 0, sinon, cette formule asymptotique est un célèbre théorème de Dirichlet datant de 1837. En particulier, Dirichlet a prouvé que pour tous a, q tels que q ≥ 1 et (a, q) = 1, il existe une infinité de nombres premiers congrus à a modulo q.

Dans la plupart des applications, la seule connaissance de la formule (2) n'est pas suffisante. On y rencontre alors deux autres importantes questions. La première concerne l'uniformité par rapport à q. La deuxième repose sur l'ordre de grandeur du terme d'erreur E f (X, q, a) := S f (X, q, a) -1 ϕ(q) S f (X, q).

(3)

Répartition de fonctions arithmétiques Donnons quelques exemples pour clarifier les idées. On reviendra plusieurs fois à ces exemples pour parler des différents problèmes liés à la répartition de f dans les progressions arithmétiques.

Exemple 1. La suite des nombres premiers. Soit P = {2, 3, 5, . . .} l'ensemble des nombres premiers. D'après le théorème de Siegel-Walfisz sur les régions sans zéro des fonctions L de Dirichlet, on sait que pour tout réel A ≥ 1, on a S P (X, q, a) = S P (X, q) ϕ(q)

+ O A X (log X) A (4) = 1 ϕ(q) X 2 dt log t + O A X (log X) A ,
uniformément pour tout X ≥ 2, pour tous a, q tels que (a, q) = 1 et 1 ≤ q ≤ X. De plus les constantes implicites ne dépendent que de A. Cela veut dire que la formule asymptotique (2) reste vraie uniformément pour q aussi grand qu'une puissance de log X et le terme d'erreur est également plus petit que le terme principal par un facteur d'une puissance de log X. L'hypothèse de Riemann généralisée pour les fonctions L de Dirichlet (notée HRG dans la suite) entraînerait, pour tout > 0, la formule asymptotique S P (X, q, a) = S P (X, q) ϕ(q)

+ O X 1 2 + , (5)
où la constante implicite ne dépend que de . On aurait donc que la formule asymptotique (2) serait valable uniformément pour q ≤ X 1 2 -, ce qui est bien meilleur que ce que l'on peut prouver inconditionnellement (voir (4)). Néanmoins, Montgomery a conjecturé quelque chose de plus fort encore que (5) :

Conjecture A. Soit > 0. Pour tout X ≥ 1 et pour tous a, q entiers tels que q ≤ X et (a, q) = 1, on a l'inégalité E P (X, q, a) X (X/q) 1 2 , où la constante implicite ne dépend que de .

Remarquons que la conjecture originale de Montgomery [START_REF] Montgomery | Topics in multiplicative number theory[END_REF]Formula (15.9)] a été infirmée par Friedlander et Granville [START_REF] Friedlander | Limitations to the equi-distribution of primes[END_REF] quand q est extrêmement proche de X. La version que nous présentons a été proposée par Friedlander et Granville dans [START_REF] Friedlander | Limitations to the equi-distribution of primes[END_REF] et répond à cette critique. Avant de passer à l'exemple suivant, on donne une définition qui nous permettra de mieux décrire le domaine de validité de la formule asymptotique (2). Définition 1. Soit f une fonction arithmétique telle que, pour tout n ≥ 1, f (n) ≥ 0. On définit l'exposant de répartition de f , dénoté θ(f ) comme étant la plus grande valeur de θ telle que pour tout > 0 et pour tout A ≥ 1 il existe C( , A) > 0 telle que, pour tout X ≥ 2, tout a et q tels que 1 ≤ q ≤ X θ-, (a, q) = 1, on ait l'inégalité On remarque que d = d 2 . Une simple application de la méthode de l'hyperbole de Dirichlet fournit que θ(d k ) ≥ 1 k . Cependant, on peut faire mieux. En effet, il a été démontré indépendamment par Selberg, Hooley et Linnik que l'on a θ(d 2 ) ≥ 2 3 . Une preuve simple de ce fait peut être obtenue à partir de la formule de Voronoi combinée avec les majorations de Weil pour les sommes de Kloosterman.

La seule autre valeur de k pour laquelle on sait que l'exposant de répartition de d k est strictement supérieur à 1 2 est k = 3, grâce au profond résultat de Friedlander et Iwaniec [START_REF] Friedlander | Incomplete Kloosterman sums and a divisor problem[END_REF] qui garantit que θ(d 3 ) ≥ 1 2 + 1 230 . Très récemment, Fouvry-Kowalski-Michel [START_REF] Fouvry | On the exponent of distribution of the ternary divisor function[END_REF] ont amélioré ce résultat en démontrant que l'on peut prendre θ(d 3 ) ≥ 1 2 + 1 46 . Pour k = 4, il aurait été connue à Linnik que θ(d 4 ) ≥ 1 2 , mais une référence précise de ce résultat semble difficile à trouver. Pour finir, lorsque k ≥ 5, le meilleur résultat est dû à Friedlander et Iwaniec [START_REF] Friedlander | The divisor problem for arithmetic progressions[END_REF]. Ils établissent que θ(d k ) ≥ θ k , avec Enfin, notre dernier exemple porte sur la suite des entiers sans facteur carré.

θ k =     
Exemple 3. La suite des entiers sans facteur carré. Soit Q l'ensemble des entiers positifs sans facteur carré. On note que Q = {n ∈ Z >0 ; µ 2 (n) = 1}, où µ est la fonction de Möbius définie de la manière suivante :

µ(n) =      1,
pour n = 1, (-1) k , si n est le produit de k premiers distincts, 0, s'il existe p premier tel que p 2 divise n.

En ce qui concerne la répartition de µ 2 dans les progressions arithmétiques, un premier résultat de Prachar [START_REF] Prachar | Über die kleinste quadratfreie Zahl einer arithmetischen Reihe[END_REF] permet d'établir, pour tout > 0, l'inégalité Théorème 1. Pour tout > 0, on a la formule asymptotique

n≤X n≡a (mod q) µ 2 (n) ∼ 1 ϕ(q) n≤X (n,q)=1 µ 2 (n)   ∼ 6 π 2 p|q 1 - 1 p 2 -1 X q   ,
lorsque X → ∞, pourvu que q soit premier, (a, q) = 1 et q ≤ X 2 3 (log X)

1 6 -.
En ce qui concerne le terme d'erreur, on peut voir grâce à la majoration de Hooley (6) que pour tout > 0, on a l'inégalité E µ 2 (X, q, a) X q

1 2 INTRODUCTION 13
uniformément pour q ≤ X 1 2 -. Le théorème suivant dit que l'on peut remplacer 1 2 par des exposants plus petits dans la majoration [START_REF] Fouvry | On the size of the fundamental solution of the Pell equation[END_REF] (se souvenir que la Conjecture B prévoit que l'on devrait même avoir l'exposant 1 4 + ).

Théorème 2. (voir Théorème 3.1.4) Pour tout η > 0, il existe δ = δ(η) > 0 tel qu'on ait l'inégalité E µ 2 (X, q, a) η X q 1 2 -δ , uniformément pour tout X ≥ 1, tout q premier tel que X η ≤ q ≤ X 1 2 -η et tout entier a premier avec q.

Si X α ≤ q ≤ 2X α avec 0 < α < 2 5 , on peut même être plus précis. Le Théorème 3.1.2 ainsi que la remarque qui le suit nous permettent d'établir le résultat suivant. Théorème 3. Pour tout α et tout vérifiant 0 < α < 2 5 et > 0, il existe C(α, ) > 0 telle que pour tout X ≥ 1, pour tout q premier vérifiant X α ≤ q ≤ 2X α et tout entier a premier avec q, on ait l'inégalité E µ 2 (X, q, a) ≤ C(α, )

X q II Majorations en moyenne sur a modulo q L'étude individuelle des termes d'erreur E f (X, q, a) est souvent ardue. En revanche un effet de moyenne sur a modulo q peut simplifier l'estimation. Considérons par exemple la variance V f (X, q) := * a (mod q) E f (X, q, a) 2 , où l'ast'erisque indique que l'on ne somme que sur les résidus a tels que (a, q) = 1. Il serait peut-être plus naturel de considérer simplement la moyenne M f (X, q) := * a (mod q) |E f (X, q, a)| .

Cependant, du point de vue analytique, la variance V f (X, q) est beaucoup plus agréable à manipuler. En général, on donne des estimations pour V f (X, q) et on en déduit une éstimation pour M f (X, q) par une application de Cauchy-Schwarz :

M f (X, q) ≤ ϕ(q) 1 2 V f (X, q) 1 2 .
Revenons à présent aux trois exemples de la partie précédente.

Majorations en moyenne sur a modulo q La suite des nombres premiers L'étude de cette suite est tellement compliquée que l'on ne peut rien prouver de satisfaisant sans supposer quelque chose de très profond comme HRG. Dans ce cadre, Túran [START_REF] Turán | Über die Primzahlen der arithmetischen Progression[END_REF] a prouvé que sous HRG, on a l'inégalité V P (X, q) X(log X) 4 , uniformément pour q ≤ X. Cela implique que le nombre d'exceptions à la conjecture de Montgomery (A) est négligeable. Plus précisément, pour tout > 0, on définit E P (X, q; ) := # a ∈ (Z/qZ) * ; |E P (X, q, a)| > X (X/q) 1 2

.

On a alors l'estimation E P (X, q) q 1-uniformément pour q ≤ X.

Les fonctions nombre de diviseurs

Pour la fonction d, des moyennes sur a modulo q ont été étudiées par plusieurs auteurs (voir par exemple [START_REF] Banks | On the average value of divisor sums in arithmetic progressions[END_REF] et [2]). Un résultat de Lau et Zhao [START_REF] Lau | On a variance of Hecke eigenvalues in arithmetic progressions[END_REF] donne en particulier une formule asymptotique pour V d (X, q) lorsque q ≥ X 1 2 + . En effet, ils établissent qu'il existe un polynôme P 3 de degré trois tel que, pour tout > 0, on ait l'égalité

V d (X, q) = P 3 log q 2 X X + O d(q)X 5 6 q 1 6 + d(q)X 5 4 q -1 4
, uniformément pour X 1 2 ≤ q ≤ X. Autrement dit, l'ordre de grandeur moyen de E d (X, q, a) est de (X/ϕ(q))

1 2 log q 2 X 3/2
. Dans le même article, Lau et Zhao prouvent que pour tout > 0, il existe δ = δ( ) > 0 telle qu'on ait V d (X, q) X 1-δ uniformément pour q ≤ X 1 2 -. Cela indique un changement de comportement lorsque q = X 1 2 rendant très difficile de conjecturer l'ordre de grandeur de E d (X, q, a) lorsque q franchit la borne X 1 2 .

INTRODUCTION 15

Remarque. D'après la conjecture de Montgomery, la majoration [START_REF] Fouvry | Gaussian distribution for the divisor function and Hecke eigenvalues in arithmetic progressions[END_REF] ne devrait être optimale pour aucune valeur de q ≤ X 1-. Remarquons tout de même qu'en modifiant très légèrement la preuve de Blomer, il est possible de remplacer le terme X 1+ par X 1 2 + q 1 2 , obtenant ainsi une estimation en accord avec la conjecture de Montgomery pour q ≥ X 7 9 + .

Dans le théorème suivant, on a repris la démarche de Blomer, mais en y apportant deux modifications. La première consiste à utiliser le crible à carrés de Heath-Brown [START_REF] Heath-Brown | The square sieve and consecutive square-free numbers[END_REF] pour obtenir des compensations grâce la somme sur dans [START_REF] Fouvry | On a theorem of Bombieri-Vinogradov type[END_REF]. Deuxièmement, on s'inspire de la technique de Croft [START_REF] Croft | Square-free numbers in arithmetic progressions[END_REF] qui avait démontré une formule asymptotique pour q≤Q V µ 2 (X, q), lorsque Q ≥ X 1 2 + , pour tout > 0. Cela nous permet d'obtenir une formule asymptotique pour V µ 2 (X, q) lorsque q est suffisamment grand par rapport à X, sans avoir besoin de la somme sur q ≤ Q. On a donc le Théorème suivant (voir Théorème 1.1.1). Théorème 4. Il existe une constante absolue C > 0 telle que, pour tout > 0, on a, uniformément pour 1 ≤ q ≤ X, l'égalité

V µ 2 (X, q) = C p|q 1 + 2p -1 -1 X 1 2 q 1 2 + O X 1 3 + q 2 3 + X 23 15 + q -13 15 ,
où la constante implicite ne dépend que de .

On en déduit alors la formule asymptotique

V µ 2 (X, q) ∼ C p|q 1 + 2p -1 -1 X 1 2 q 1 2 , lorsque X → ∞, avec X 31 41 + ≤ q ≤ X 1-.
Nous signalons que Le Boudec [START_REF] Boudec | On the distribution of squarefree integers in arithmetic progressions[END_REF] a récemment démontré une majoration du bon ordre de grandeur pour V µ 2 (X, q) lorsque q ≥ X 1 2 + , pour tout > 0. Sa preuve s'appuie sur des résultats de géométrie des nombres.

III Une question de dépendance de variables aléatoires

Une question très intéressante qui a été étudiée dans le cadre de la fonction d par Fouvry et al. [START_REF] Fouvry | Gaussian distribution for the divisor function and Hecke eigenvalues in arithmetic progressions[END_REF] est celle de la corrélation entre les variables aléatoires a → E f (X, q, a)

(V f (X, q)/ϕ(q)) 1/2 et a → E f (X, q, γ(a)) (V f (X, q)/ϕ(q)) 1/2 , (11)
pour une fonction γ : Z/qZ → Z/qZ de nature algébrique. Plus précisément, ils se sont intéressés au cas où q est premier et γ est une fonction homographique de la droite projective

P 1 (F q ). C'est-à-dire γ(a) = ra + s ta + u , (a = tu) (12)
Une question de dépendance de variables aléatoires avec r, s, t, u ∈ Z, ∆ γ := ru -st = 0, où l'on note x l'inverse multiplicatif d'un résidu primitif x modulo q. On a étudié ce problème dans le cadre de la fonction µ 2 pour γ une fonction affine, i.e. γ : Z/qZ → Z/qZ donnée par

γ(a) = ra + s (13)
Cela correspond à la définition [START_REF] Fouvry | On the exponent of distribution of the ternary divisor function[END_REF] dans le cas particulier où t = 0 et u = 1. Le Théorème 5 ci-dessous résume nos résultats sur ce sujet. Pour toute fonction γ comme en [START_REF] Fouvry | On the exponent of distribution of the ternary divisor function[END_REF], on définit la somme de covariance

C[γ](X, q) := a (mod q) a =0,-rs,-tu E µ 2 (X, q, a)E µ 2 (X, q, γ(a)). ( 14
)
On a le résultat suivant, qui combine les Théorèmes 1.1.3 et 2.1.2.

Théorème 5. Soit γ une fonction affine comme en (13) et soit C[γ](X, q) comme en [START_REF] Friedlander | Limitations to the equi-distribution of primes[END_REF]. Il existe des constantes C γ et δ > 0 telles que pour tout > 0, on ait l'égalité

C[γ](X, q) = C γ X 1 2 q 1 2 + O ,r X 1 2 q 1 2 (log X) -δ
uniformément pour X ≥ 2 et pour q premier tel que q > |r|, q s, et X 7 9 + ≤ q ≤ X 1-, où la constante implicite ne dépend que de et r. De plus, si s est différent de 0, alors C γ = 0.

Nous traitons les cas s = 0 et s = 0 séparément. Si s = 0, alors la preuve suit les mêmes lignes que celles employées dans le cadre du Théorème 4 et que nous avons décrites dans la partie précédente. Si s = 0, le fait que C γ = 0 est une conséquence d'une majoration de sommes d'exponentielles due à Bourgain (voir Théorème F ci-dessous).

Nous croyons que les méthodes ci-dessus s'étendent à l'étude de C[γ](X, q) pour toute fonction γ comme en (12) avec t = 0. De plus, le cas où γ est telle que t = 0, un argument se basant sur la majoration de Weil sur les sommes de Kloosterman classiques devrait donner un résultat analogue au Théorème 5 avec C γ = 0 pour tout γ avec t = 0.

Le Théorème 5 peut être interprété de la manière suivante. Soit (X n ) n≥1 une suite croissante de réels et (q n ) n≥1 une suite croissante de nombres premiers tels que X n , q n → ∞ avec la contrainte

X 7 9 + n ≤ q n ≤ X 1-
n . Alors le coefficient de corrélation entre les suites de variables aléatoires

a → E f (X n , q n , a) et a → E f (X n , q n , γ(a))
converge, lorsque q tend vers l'infini, vers -une constante C r = 0 si s = 0. Donc ces variables ne sont pas asymptotiquement indépendantes. -0 si s = 0. Cela suggère que ces variables peuvent être asymptotiquement indépendantes.

Mais, il nous faudrait plus d'informations sur les moments mixtes d'ordre supérieur comme dans [START_REF] Fouvry | Gaussian distribution for the divisor function and Hecke eigenvalues in arithmetic progressions[END_REF] et [START_REF] Kowalski | Fourier coefficients of GL(N ) automorphic forms in arithmetic progressions[END_REF] pour pouvoir établir l'indépendance.

IV Moments d'ordre supérieur

Nous pouvons aussi considérer les moments d'ordre supérieur

M f (X, q; ) = * a (mod q) E f (X, q, a) . (15)
Dans le cas où f = d, Fouvry et al. ont démontré, pour tout ≥ 2 et tout > 0, une formule asymptotique valable uniformément pour X

1 2 + ≤ q ≤ X 2 -1 -.
Comme conséquence ils ont obtenu une loi de répartition gaussienne pour E d (X, q, a) lorsque q est proche de X 1 2 .

Nous nous sommes inspirés de ce résultat pour étudier les moments M f (X, q; ) dans le cas où f = µ 2 . Hall [START_REF] Hall | The distribution of squarefree numbers[END_REF] avait considéré le problème des moments pour la répartition des entiers sans facteur carré dans les petits intervalles. Il existe une analogie entre la répartition de suites arithmétiques dans les progressions arithmétiques et dans les petits intervalles. Nous développons cette analogie dans le cas de la fonction µ 2 dans le Chapitre 4. En particulier les résultats de Hall se traduisent dans le contexte des progressions arithmétiques de la faà §on suivante :

Théorème C. (Hall) Soit ≥ 2 un entier et soit M µ 2 (X, q; ) comme en (15) avec f = µ 2 . Alors il existe δ( ) > 0 telle que M µ 2 (X, q; ) , ϕ(q) X q -1 2
, uniformément pour X ≥ 1 et X 1-δ( ) ≤ q ≤ X 1-, où la constante implicite ne dépend que de et de .

Si la Conjecture B est vraie, on pourrait remplacer le second membre de l'inégalité du Théorème C par ϕ(q) (X/q) 4 + . La méthode de Hall est assez différente de celle de Fouvry et al. La dernière se base sur une application de la formule de Voronoi combinée avec des estimations de sommes de corrélations liées aux sommes de Kloosterman (il s'agit d'un théorème assez profond d'indépendance de groupes de monodromie liés à ces sommes).

Dans le cas de la fonction µ 2 , nous n'avons pas d'équivalent de la formule de Voronoi (Les fonctions L associées ont une infinité de pôles dans la bande 0 < (s) < 1 4 ). Il reste l'alternative naïve : développer le terme E µ 2 (X, q, a) et changer l'ordre de sommation. Nous devons alors étudier, pour 0 ≤ j ≤ , les termes de la forme

T ,j (X, q) := (-1) -j j S µ 2 (X, q) ϕ(q) -j . . . * 0<n 1 ,...,n j ≤X n 1 ≡...≡n j (mod q) µ 2 (n 1 ) . . . µ 2 (n j ), (16)
où S µ 2 (X, q) est comme défini dans [START_REF] Banks | On the average value of divisor sums in arithmetic progressions[END_REF]. Il est facile de donner un équivalent asymptotique de chaque T ,j (X, q). On parvient alors à la majoration

M µ 2 (X, q; ) ϕ(q) X q -3 2 ,
Moments d'ordre supérieur ce qui est moins bon que le Théorème C pour tout ≥ 3. Il est donc impératif d'étudier la contribution simultanée des termes dans [START_REF] Friedlander | Incomplete Kloosterman sums and a divisor problem[END_REF]. Cela se fait grâce à la proposition suivante (voir par exemple l'équation (4.95)).

Proposition 6. Soit ψ : R → -1 2 , 1 2 la fonction définie par

x = x - 1 2 + ψ(x).
Soit ≥ 3 un entier et soit M µ 2 (X, q; ) comme en [START_REF] Friedlander | The divisor problem for arithmetic progressions[END_REF]

avec f = µ 2 . Soit C : R >0 × Z >0 → R donnée par C (Y, q) = ∞ r 1 =1 (r 1 ,q)=1 . . . ∞ r =1 (r ,q)=1 µ(r 1 ) . . . µ(r ) r2 E (Y ; r 1 , . . . , r ), (17)
où r = ppcm[r 1 , . . . , r ] et E (Y ; r 1 , . . . , r ) := r2 0 j=1 ψ Y -u r 2 j -ψ -u r 2 j du.
Alors, il existe δ( ) > 0 telle que pour tout > 0 et tout q entier vérifiant X 1-δ( ) ≤ q ≤ X 1-, on ait l'égalité

1 ϕ(q) M µ 2 (X, q; ) = C X q , q + O (X -1 100 ).
La Proposition 6 réduit le problème des moments M µ 2 (X, q; ) à une étude du comportement asymptotique de la fonction C , du moins pour q assez grand par rapport à X. En particulier, les méthodes de Hall fournissent l'inégalité C X q , q X q -1 2

.

Pour ce faire, Hall fait appel au lemme fondamental de Montgomery et Vaughan (voir [START_REF] Montgomery | On the distribution of reduced residues[END_REF]Lemma 1]). Nous remarquons que la fonction C (k) définie par Hall dans [START_REF] Hall | The distribution of squarefree numbers[END_REF] semble être assez différente de celle définie dans [START_REF] Friedlander | Opera de cribro[END_REF], mais nous verrons dans le Chapitre 4 que ces deux fonctions peuvent être traitées de la même faà §on. Quand = 2, on démontre que

E (Y ; r 1 , r 2 ) = 2d 2 Y d 2 0 ψ(v)dv ≤ min d 2 , Y , (18)
où d = pgcd(r 1 , r 2 ) et cela suffit pour obtenir une majoration du bon ordre de grandeur pour V µ 2 (X, q) = M µ 2 (X, q; 2). Pour ≥ 3 nous n'avons pas trouvé d'analogue à [START_REF] Hall | Squarefree numbers on short intervals[END_REF]. Nous avons tout de même une majoration de E (Y ; r 1 , . . . , r ), qui s'écrit essentiellement comme INTRODUCTION 19 Théorème 7. Soit ≥ 3 un entier et soit M µ 2 (X, q; ) comme en (15) avec f = µ 2 . Soit θ donnée par

θ :=          2 3 + 2 , for = 3, 4, - 2 2
, for ≥ 5.

Soit δ donnée par

δ := 1 ( + 1)( -θ ) .
Alors pour tout > 0 et tout ≥ 3 il existe C , > 0 tel que pour tout X ≥ 1 et tout entier positif q ≤ X, on ait l'inégalité

M µ 2 (X, q; ) ≤ C , ϕ(q) X q θ + + X +1 X q -1
.

En particulier, pour tout > 0, pour tout ≥ 3, pour tout X ≥ 1 et tout entier positif q tel que X 1-δ + ≤ q ≤ X, on a l'inégalité

M µ 2 (X, q; ) ≤ 2C , ϕ(q) X q θ + .
En particulier, pour tout ≥ 3, on a θ < -1 2 (comparer avec le Théorème C). Selon la Conjecture B, les valeurs de θ dans le Théorème 7 ne devraient pas être optimales. En fait, la Conjecture B et les résultats de [START_REF] Fouvry | Gaussian distribution for the divisor function and Hecke eigenvalues in arithmetic progressions[END_REF] pour la fonction d nous permettent d'espérer que la conjecture suivante soit vraie : Conjecture 8. Pour tout entier ≥ 1 et tout > 0, il existe δ > 0, η > 0, X 0 ( , ) > 0 et une fonction positive et multiplicative c tels qu'on ait l'inégalité

M µ 2 (X, q; ) -c (q)ϕ(q) X q 4 ≤ ϕ(q) X q 4 -η , (19)
pour tous q et X tels que

X 1-δ ≤ q ≤ X 1-, X ≥ X 0 ( , ).
De plus -si est pair, la fonction q → c (q) est bornée, -si est impair, c (q) = 0 pour tout q.

Remarquons que pour = 1 cette conjecture est triviale, vu que le membre de gauche vaut 0, et pour = 2, la conjecture est vraie avec η 2 < 1 6 et δ 2 > 18 23 grâce au Théorème 4.

V Sommes d'exponentielles

Une partie cruciale des résultats des Chapitres 2 et 3 vient de majorations de sommes d'exponentielles. Nous voulons donc dans cette partie faire un panorama des résultats utilisés.

Sommes d'exponentielles

Majorations classiques

La majoration suivante est une conséquence des travaux de Weil sur l'hypothèse de Riemann pour les courbes sur les corps finis. Rappelons que e(x) := exp(2πix).

Théorème D. On a l'inégalité

q * n=1 e an + bn 2 q ≤ 3q 1 2 (q, a, b) 1 2
pour tous a et b entiers et q premier suffisamment grand.

Sommes d'exponentielles (très) courtes

Le Théorème D donne une excellente majoration pour la somme complète (sur tous les résidus primitifs modulo q). Par la méthode de complétion, on accède à une majoration nontriviale pour toutes les sommes de la forme

N * n=M +N e an + bn 2 q (20)
dès que N ≥ q 1 2 log q. Cependant, pour quelques applications, nous avons besoin d'estimations non-triviales pour des sommes avec N plus petit. Karatsuba [START_REF] Karatsuba | The distribution of inverses in a residue ring modulo a given modulus[END_REF] a étudié des sommes de ce type avec M = 0 et N très petit (N ≥ q , pour fixé). La version que nous donnons ici est due à Bourgain et Garaev [5].

Théorème E. On a l'inégalité * n≤N e an q N log N (log log N ) 3 (log q) 3 2
uniformément sur N ≥ 3, q premier et a entier avec (a, q) = 1.

Remarque. La condition M = 0 (voir ( 20)) est essentielle ici, vu que la technique utilisée dépend fortement de la factorisation des entiers sommés.

INTRODUCTION 21

Sommes d'exponentielles tordues par la fonction µ

Dans le Chapitre 3 on considère des sommes d'exponentielles tordues par la fonction µ pour lesquelles nous avons besoin d'estimations non-triviales. On présente ici les deux résultats principaux dont nous nous servons. Ici, comme dans plusieurs théorèmes en théorie analytique de nombres, les techniques permettant d'étudier ces sommes permettent aussi de traiter les sommes sur les nombres premiers. Pour tout N ≥ 1, tout q entier et pour tout a premier avec q, on considère les sommes d'exponentielles suivantes :

Σ µ (N ; q, a) := n≤N µ(n)e an 2 q et Σ P (N ; q, a) := p≤N e ap 2 q , (21)
où dans Σ P (N ; q, a) on somme sur les nombres premiers p ≤ N . La première estimation que nous utilisons est un cas très particulier de [START_REF] Fouvry | Algebraic trace functions over the primes[END_REF]Theorem 1.7].

Théorème G. Soit Σ µ (N ; q, a) comme dans [START_REF] Heath-Brown | The least square-free number in an arithmetic progression[END_REF]. Alors pour tout > 0 on a l'inégalité

Σ µ (N ; q, a) N 1 + q N 1 12 q -1 48 +
uniformément pour N ≥ 1, q premier et a premier avec q. La constante implicite ne dépend que de .

Le Théorème G donne une majoration non-triviale pour tout N ≥ q 3 4 + et nous pouvons quantifier le gain par rapport à la borne triviale. Dans [3], Bourgain présente une méthode permettant d'avoir des majorations non triviales pour N ≥ q 1 2 + mais où le gain n'est pas explicite. Le théorème suivant est une conséquence de [3,Theorem A.9].

Théorème H. Soit Σ µ (N ; q, a) comme dans [START_REF] Heath-Brown | The least square-free number in an arithmetic progression[END_REF]. Alors pour tout > 0, il existe δ = δ( ) > 0 telle qu'on ait, uniformément pour q entier, q 1 2 + ≤ N ≤ q et a premier avec q, l'inégalité Σ µ (N ; q, a) N 1-δ , où la constante implicite ne dépend que de .

Remarque. Comme dans la sous-section précédente, les Théorèmes G et H utilisent fortement la factorisation des n sur lesquels on somme et donc la méthode n'est valable essentiellement que pour des sommes commenà §ant en 0.

Remarque. Les Théorèmes G et H restent vrais avec Σ P (N ; q, a) à la place de Σ µ (N ; q, a).

Chapitre 1

Squarefree numbers in arithmetic progressions 1

Introduction

For a positive real number X and positive integers a, q with (a, q) = 1, let E(X, q, a) be defined by the formula

n≤X n≡a (mod q) µ 2 (n) = 6 π 2 p|q 1 - 1 p 2 -1 X q + E(X, q, a), (1.1)
where, as usual, µ is the Möbius function. In (1.1) the first term heuristically appears to be a good approximation to the number of squarefree integers ≤ X congruent to a (mod q). In this paper, we are concerned with the so called error term E(X, q, a). Trivially, one has

|E(X, q, a)| ≤ X q + 1, (1.2)
while in [START_REF] Hooley | A note on square-free numbers in arithmetic progressions[END_REF], Hooley proved

E(X, q, a) = O X q 1 2 + q 1 2 + , (1.3)
where the O -constant depends only on > 0 arbitrary. This is the best result available for fixed a. Furthermore, (1.3) gives an asymptotic formula for the left-hand side of (1.1) for q ≤ X 2 3 -. The same range of validity for this asymptotic formula had already been established by Prachar [START_REF] Prachar | Über die kleinste quadratfreie Zahl einer arithmetischen Reihe[END_REF], with a weaker error term than (1.3). We believe that such an asymptotic formula should hold for q ≤ X 1-and it is a challenging problem to go beyond X 2 3for a general q, in particular when q is prime. The situation is quite similar to that of the analogous problem for the divisor function. By this we mean that an asymptotic formula is known for n≤X n≡a (mod q) d(n) in the range q ≤ X 2 3 -, (a, q) = 1 and we do not know how to overcome this threshold for general q (in particular when q is prime). And in this case it is also strongly expected that such an asymptotic formula should hold for q ≤ X 1-.

It is in general easier to study the error terms on average. Blomer [2] considered the second moment of the E(X, q, a)

V µ 2 (X, q) := * a (mod q) |E(X, q, a)| 2 , (1.4)
where the * symbol means that we only sum over the classes that are relatively prime to q. In [2, Theorem 1.3], he showed that

V µ 2 (X, q) X X + min X 5 3 q , q 2 (1.5)
holds for every > 0, uniformly for 1 ≤ q ≤ X. Several years before, Croft [START_REF] Croft | Square-free numbers in arithmetic progressions[END_REF] considered a variation of V µ 2 (X, q) by summing not only over the classes relatively prime to q but over all classes (mod q). Let

V µ 2 (X, q) := a (mod q)        n≤X n≡a (mod q) µ 2 (n) - µ 2 (d)q 0 ϕ(q 0 ) 6 π 2 p|q 1 + p -1 -1 X q        2 ,
where d = (a, q) and q 0 = q/d. The last term between the curly brackets on the expression above can be seen as the expected value of the sum n≤X n≡a (mod q) µ 2 (n) when a is not necessarily relatively prime to q. Observe that it reduces to the first term on the right-hand side of (1.1) whenever (a, q) = 1. By taking an extra average over q, Croft [7, Theorem 2] proved the following formula q≤Q V µ 2 (X, q) = BX

1 2 Q 3 2 + O X 2 5 Q 8 5 log 13 5 X + X 3 2 log 7 2 X , (1.6)
uniformly for Q ≤ X, where B is an explicit constant. For Q ≥ X

V µ 2 (X, q)
which is an analogue of the classical Barban-Davenport-Halberstam Theorem with a main term (see Montgomery [START_REF] Montgomery | Primes in arithmetic progressions[END_REF]). This result was further improved by several authors. We mention the Introduction works of Warlimont [START_REF] Warlimont | Squarefree numbers in arithmetic progressions[END_REF], Brüdern et al. [START_REF] Brüdern | On the exponential sum over k-free numbers[END_REF] and Vaughan [START_REF] Vaughan | A variance for k-free numbers in arithmetic progressions[END_REF]. It is worth mentioning that the later two results deal with the more general case of k-free numbers.

Ignoring for the moment the difference between V µ 2 (X, q) and V µ 2 (X, q), formula (1.6) above can be interpreted as saying that, at least on average over q ≤ Q, (1.5) is far from the truth. In this paper we investigate if such a phenomenon can be observed without the need of the extra average over q. This is indeed possible for q large enough relative to X. Our main theorem goes in this direction Theorem 1.1.1. Let V µ 2 (X, q) be defined as in (1.4) and > 0 arbitrary. Then, uniformly for q ≤ X, we have

V µ 2 (X, q) = C p|q 1 + 2p -1 -1 X 1 2 q 1 2 + O X 1 3 + q 2 3 + X 23 15 + q -13 15 ,
where

C = ζ 3 2 π p p 3 -3p + 2 p 3 = 0.167 . . . (1.7)
and the O -constant depends at most on .

As pointed out to the author by Zeev Rudnick, the same constant was found by Hall [START_REF] Hall | Squarefree numbers on short intervals[END_REF] on the main term of the second moment for the problem of squarefree numbers in short intervals. Notice that for X 8 13 + < q < X 1-, Theorem 1.1.1 improves upon the upper bound (1.5). We have further the following direct consequence of Theorem 1.1.1 Corollary 1.1.2. Let V µ 2 (X, q) be defined as in (1.4) and > 0 arbitrary. Then we have

V µ 2 (X, q) ∼ C p|q 1 + 2p -1 -1 X 1 2 q 1 2
(1.8) as X, q → ∞, with q satisfying X 31 41 + ≤ q ≤ X 1-, where C is as in (1.7).

Remark 1.1.1. Asymptotic formula (1.8) gives an average order of magnitude of O (X/q) 1 4 + for the terms E(X, q, a). This remark goes in the direction of a conjecture due to Montgomery (see [7, top of the page 145]), which we write under the form

E(X, q, a) = O (X/q) 1 4 +
, > 0 arbitrary uniformly for (a, q) = 1, X θ 1 < q < X θ 2 where the values of the constants θ 1 and θ 2 satisfying 0 < θ 1 < θ 2 < 1 have to be made precise.

Note that the same phenomena can be observed in the work of Croft, with an extra average over q. Such a conjecture may, of course, be interpreted in terms of the poles of the functions L(χ,s) L(χ 2 ,2s) , where χ is a Dirichlet character modulo q.

1.1.1 -Discussion about Γ an (m) and Γ ar (m)

25

Our technique is more general since we can also shed a light on the correlation between E(X, q, a) and E(X, q, ma) for fixed m with (m, q) = 1. For X > 0 and integers m and q such that m = 0, (m, q) = 1, q ≥ 1 we define C[m](X, q) = * a (mod q) E(X, q, a)E(X, q, ma).

(1.9) Theorem 1.1.1 above can be deduced as a consequence of the following Theorem 1.1.3. Let m be a squarefree integer of arbitrary sign and > 0 be arbitrary. Also let C[m](X, q) be defined as in (1.9). Then, uniformly for q ≤ X, (m, q) = 1 we have

C[m](X, q) = C 2 Γ an (m)Γ ar (m) p|q 1 + 2p -1 -1 X 1 2 q 1 2 + O ,m X 1 3 + q 2 3 + X 23 15 + q -13 15
,
where C is as in (1.7), the analytic factor Γ an (m) is defined by

Γ an (m) :=            √ m + 1 - √ m -1 m , if m > 0, √ 1 -m - √ -m -1 -m , if m < 0, (1.10)
and the arithmetic factor Γ ar (m) is

Γ ar (m) := p|m 1 + p + p 1 2 + 1 p 3 2 + p 1 2 + 1 -1
;

(1.11)

and the O ,m -constant depends at most on and m.

We restrict to the case where m is squarefree to avoid further complications, but we expect the proof to work for general m. It also follows from the proof that the dependency of the error term on m is polynomial.

The error term O(X 23/15+ q -13/15 ) comes from the use of the square sieve (see [START_REF] Heath-Brown | The square sieve and consecutive square-free numbers[END_REF]). The proof could be simplified by avoiding the use of this sieve, obtaining a worse error term. One can obtain O(X 5/3+ q -1 ) in a rather elementary way. That would imply that the asymptotic formula (1.8) would only hold for X 7/9+ ≤ q ≤ X 1-. Finally, notice that making the choice m = 1, one has Γ an (1) = 2, Γ ar (1) = 1 and we retrieve Theorem 1.1.1. Introduction are not asymptotically independent as X, q → ∞, with q satisfying X 31 41 + ≤ q ≤ X 1-. This is a consequence of the fact that these random variables have asymptotic mean equal to zero (see lemma 1.3.1 below). The fact that X and X m are dependent (when m > 0) can be guessed similarly as in [START_REF] Fouvry | Gaussian distribution for the divisor function and Hecke eigenvalues in arithmetic progressions[END_REF]Remark 1.8], by the trivial fact that if a squarefree n satisfies n ≡ a (mod q), 1 ≤ n ≤ X/m, (n, m) = 1, then n = mn satisfies n squarefree, n ≡ ma (mod q), 1 ≤ n ≤ X.

Such an interpretation obviously fails when m < 0 or m is not squarefree, which may explain the signs of Γ an (m). We also remark that the random variable 1 ϕ(q) m (mod q) * XX m has asymptotic mean zero, again by lemma 1.3.1.

Finally we would like to point out some differences between Theorem 1.1.3 and [9, Theorem 1.5] from which this study was inspired. In [9, Corollary 1.7], the correlation for the divisor function exists if and only if m > 0, with a correlation coefficient that is always positive, while in our setting, the correlation always exists and the sign might depend on m.

A double sum over squarefree integers

Developing the squares in C[m](X, q), we obtain the equality

C[m](X, q) = S[m](X, q) -2C(q) X q n≤X (n,q)=1 µ 2 (n) + ϕ(q) C(q) X q 2 , (1.12)
where

C(q) = 6 π 2 p|q 1 - 1 p 2 -1 , (1.13)
and S[m](X, q) is the double sum

S[m](X, q) = n 1 ,n 2 ≤X (n 1 n 2 ,q)=1 mn 1 =n 2 (mod q) µ 2 (n 1 )µ 2 (n 2 ).
(1.14)

Thus an important step in proving Theorem 1.1.1 is providing an asymptotic formula for S[m](X, q), which is the subject of our next Theorem.

Theorem 1.1.4. Let X > 2 be a real number and > 0 be arbitrary. Let m be a squarefree integer of arbitrary sign. Then, uniformly for q ≤ X, (m, q) = 1, we have

1.1.2 -A double sum over squarefree integers S[m](X, q) = ϕ(q) C(q)X q 2 + C 2 Γ an (m)Γ ar (m) p|q 1 + 2p -1 -1 X 1 2 q 1 2 + O ,m X 1 3 + q 2 3 + X 23 15 + q -13 15 ,
where C, C(q) are as in (1.7) and (1.13) respectively, Γ an (m) and Γ ar (m) are as in (1.10) and (1.11) respectively ; and the implied O ,m -constant depends at most on and m.

The notation used throughout the paper is given in section 1.2. In section 1.3 we prove that Theorem 1.1.4 implies Theorem 1.1.3. In section 1.4 we break up S[m](X, q) (see (1.14) above) into two sums S ≤y [m](X, q) and S >y [m](X, q), and begin the study of the first of these sums. In section 1.5 we prove some lemmas that prove to be useful in the following section, where we evaluate the main term. Section 1.7 is dedicated to the estimation of S >y [m](X, q) by means of the square sieve. In this section, we use some upper bounds for exponential sums that are proven in section 1.9. Finally, the proof of Theorem 1.1.4 is given in section 1.8.

Notation

For a finite set S, #S denotes its cardinality. Let ω be the well-known function defined by

ω(n) = #{p prime ; p | n},
for every integer n ≥ 1. For N > 0, we use the notation n ∼ N to mean N < n ≤ 2N . Let ψ(v) denote the sawtooth function defined by

v = v - 1 2 + ψ(v). (1.15) For n ∈ Z, n = 0 we define σ(n) = p 2 |n p. (1.16) If A ⊂ R and α ∈ R, then αA := {αx ; x ∈ A} .
For a measurable set A ⊂ R, |A| denotes its Lebesgue measure.

1.3 Proof of Theorem 1.1.3 assuming Theorem 1.1.4

We start with the following lemma, whose proof follows closely that of [START_REF] Hardy | An introduction to the theory of numbers[END_REF]Theorem 333] Lemma 1.3.1. Let > 0 be given. We then have the equality

n≤X (n,q)=1 µ 2 (n) = ϕ(q) q 6 π 2 p|q 1 - 1 p 2 -1 X + O X 1 2 +
, uniformly for 1 ≤ q ≤ X, where the O -constant depends on alone.

Initial Steps

By Lemma 1.3.1 and (1.12), we have the equality

C[m](X, q) = S[m](X, q) -ϕ(q) C(q) X q 2 + O(X 3 2 + q -1 ).
Now, we use Theorem 1.1.4 for the term S[m](X, q) and the main terms disappear. We deduce

C[m](X, q) = C 2 Γ an (m)Γ ar (m) p|q 1 + 2p -1 -1 X 1 2 q 1 2
+ O ,m (X

1 3 + q 2 3 + X 23 15 + q -13 15 + X 3 2 + q -1 ),
and the second error term dominates the last one. This concludes the proof of Theorem 1.1.3.

Initial Steps

We are considering the sum

S[m](X, q) = n 1 ,n 2 ≤X (n 1 n 2 ,q)=1 mn 1 =n 2 (mod q) µ 2 (n 1 )µ 2 (n 2 ).
We put q = n 2 -mn 1 and write S[m](X, q) as

S[m](X, q) = ∈Z n∈I( ) (n,q)=1 µ 2 (n)µ 2 (mn + q), (1.17)
where I( ) is the interval defined by

I( ) =    (0, X) ∩ -q m , X-q m if m > 0 (0, X) ∩ X-q m , -q m if m < 0. (1.18) Note that since µ 2 (n)µ 2 (mn+h) = 1 if and only if σ(n)σ(mn+h) = 1 (recall definition (1.16)), we have µ 2 (n)µ 2 (mn + h) = d|σ(n)σ(mn+h) µ(d). (1.19)
We now use formula (1.19) with h = q in equation (1.17) and change the order of summation.

We thus obtain

S[m](X, q) = ∈Z 1≤d≤X (d,q)=1 µ(d)N d ( ), (1.20)
where N d ( ) := #{n ∈ I( ); (n, q) = 1 and σ(n)σ(mn + q) ≡ 0 (mod d)}.

(1.21)

1.1.2 -A double sum over squarefree integers 29 Let 0 < y < X be a parameter to be chosen later depending on X and q. We break up the sum in (1.20) as follows

S[m](X, q) = S ≤y [m](X, q) + S >y [m](X, q), (1.22)
where

             S ≤y [m](X, q) = ∈Z d≤y (d,q)=1 µ(d)N d ( ), S >y [m](X, q) = ∈Z y<d≤X (d,q)=1 µ(d)N d ( ).
(1.23)

Lemma 1.4.1. Let m = 0 be a squarefree number and y > 1. Let S ≤y [m](X, q) be defined by (1.23). Then we have

S ≤y [m](X, q) = ∈Z f q ( , m) |I( )| + O m X q d(q)y + Xy -1 log y ,
uniformly for X, y > 1 and q ≥ 1 satisfying (m, q) = 1, where

f q ( , m) = C 2 p|m p 2 -1 p 2 -2 p|q p 2 -p p 2 -2 κ(( , m 2 )) p 2 | p mq p 2 -1 p 2 -2 , (1.24)
C 2 = p 1 - 2 p 2 , (1.25)
κ is the multiplicative function defined by

κ(p α ) =            p 2 -p -1 p 2 -1 , if α = 1, p 2 -p p 2 -1 , if α = 2, 0, if α ≥ 3, (1.26)
and the implied constant depends at most on m.

Démonstration. Let u p ( ) := #{v (mod p 2 ); v ≡ 0 (mod p 2 ) or mv + ≡ 0 (mod p 2 )}.

(1.27)

Notice that if (p, r) = 1, then u p ( r) = u p ( ).

Using Möbius inversion, we have that for every squarefree d coprime to q,

N d ( ) = d |q µ(d ) N d,d ( ), (1.28)
where

N d,d ( ) = # n ∈ 1 d I( ); σ(d n )σ(d (mn + q )) ≡ 0 (mod d)

Initial

Steps and q = q/d . Since we have (d, d ) = 1, then

N d,d ( ) = # n ∈ 1 d I( ); σ(n )σ(mn + q ) ≡ 0 (mod d) .
By the Chinese remainder theorem and (1.27) (notice that (p, q ) = 1, whenever p | d), we have

N d,d ( ) = U d ( ) d 2 |I( )| d + O(U d ( )), (1.29)
where

U d ( ) := p|d u p ( ).
Injecting (1.29) in (1.28), we deduce

N d ( ) = ϕ(q) q |I( )| U d ( ) d 2 + O (d(q)U d ( )) . (1.30) Notice further that if (p, m) = 1, then u p ( ) ≤ 2.
Hence we have the upper bound

U d ( ) m 2 ω(d) . (1.31)
From (1.30) and the upper bound (1.31), we deduce

d≤y (d,q)=1 µ(d)N d ( ) = ϕ(q) q |I( )| d≤y (d,q)=1 µ(d)U d ( ) d 2 + O m (d(q)y log y) = ϕ(q) q p q 1 - u p ( ) p 2 |I( )| + O m d(q)y log y + Xy -1 log y , (1.32)
where in the second line we used the trivial bound |I( )| ≤ X as well as the convergence of the infinite product appearing. Observe that if | | > (|m| + 1)X q , the set I( ) is empty and hence N d ( ) = 0 for every d. Combining this observation and equation (1.32), we deduce

S ≤y [m](X, q) = | |≤ (|m|+1)X q d≤y (d,q)=1 µ(d)N d ( ) = ϕ(q) q | |≤ (|m|+1)X q p q 1 - u p ( ) p 2 |I( )| + O m X q d(q)y + Xy -1 log y = ϕ(q) q ∈Z p q 1 - u p ( ) p 2 |I( )| + O m X q d(q)y + Xy -1 log y . (1.33)
We finish by computing of u p ( ) for every p q. We distinguish five different cases.

-

If p | m, p 2 | , then u p ( ) = p; 1.1.2 -A double sum over squarefree integers 31 -If p | m, p | but p 2 , then u p ( ) = p + 1; -If p | m, p , then u p ( ) = 1; -If p m, p 2 | , then u p ( ) = 1; -If p m, p 2 , then u p ( ) = 2.
The proof now follows from (1.33) and the different values of u p ( ).

Using Lemma 1.4.1 for S ≤y [m](X, q) in (1.22), we obtain S[m](X, q) = A[m](X, q) + S >y [m](X, q) + O m X q d(q)y + Xy -1 log y , (1.34)
where

A[m](X, q) = ∈Z f q ( , m)|I( )|. (1.35)

Preparatory results

In the next section we evaluate the sum A[m](X, q). But first we prove some preliminary results that shall be useful. We start with a lemma that is a simplified version of [7, Lemma 1]. Lemma 1.5.1. For X > 1, 0 < s < 2, and ψ(v) defined by (1.15),

X 0 ψ(v)v -s 2 dv = ζ( s 2 -1) ( s 2 -1) + O X -s 2 ,
where the O-constant is absolute.

Démonstration. We have the formula (see [36,

equation (2.1.6)]) ζ s 2 -1 = s 2 -1 ∞ 0 ψ(v)v -s 2 dv, (0 < σ < 2),
where σ denotes the real part of s. We estimate the tail of the integral

I = ∞ X ψ(v)v -s 2 dv. (1.36)
Let Ψ(v) be given by Preparatory results

Ψ(v) := v 0 ψ(u)du. (1.37)
Integration by parts gives

I = Ψ(X)X -s 2 + s 2 ∞ X Ψ(v)v -s 2 -1 dv.
Notice that since 1 0 ψ(v)dv = 0, Ψ is periodic and thus bounded. Hence, we have

I X -s 2 ,
which concludes the proof.

Let h(d) be the multiplicative function defined by

h(d) := µ 2 (d) p|d 1 -2p -2 -1 . (1.38)
This function appears naturally when we study A[m](X, q) (see (1.35)). The next lemma expresses the function h(d) as a convolution between the identity and a function of rapid decay.

Lemma 1.5.2. Let h(d) be as in (1.38). Then we have

h(d) = d 1 d 2 =d β(d 1 ),
where β(t) is supported on cubefree numbers. Furthermore, for every cubefree integer t, if we write t = ab 2 with a, b squarefree, (a, b) = 1, we have

β(t) d(a) a 2 .
Démonstration. It is easy to calculate the values of β on prime powers

β(p k ) =            1, if k = 0, 2 p 2 -2 , if k = 1, -p 2 p 2 -2 , if k = 2, 0, otherwise.
The lemma now follows because the product

p p 2 p 2 -2 is convergent.

-A double sum over squarefree integers 33

The next lemma is based on [7, Lemma 3]. However, there are some confusing steps in the corresponding proof as it was also pointed out by Vaughan (see [39, page 574]). In particular, the value for the constant B (see [7, equation 6.3]) is wrong.

Lemma 1.5.3. For Y > 0 and r an integer ≥ 1, let G(Y, r) := (d,r)=1 h(d)Ψ Y d 2 , (1.39)
where Ψ(v) is as in (1.37) and h(d) by (1.38). We have, uniformly for Y ≥ 1 and r = 0,

G(Y, r) = C p|r 1 + p(p 2 -2) -1 -1 Y 1 2 + O d(r)Y 1 3 , (1.40)
where C is given by

C = ζ 3 2 2π p p 3 -3p + 2 p(p 2 -2) , (1.41)
and the O-constant is absolute.

Démonstration. We first prove the related formula

(d,r)=1 Ψ Y d 2 = ϕ(|r|) |r| ζ( 3 2 ) 2π Y 1 2 + O d(r)Y 1 3
.

(1.42)

Let D = D(Y ) > 0 to be chosen later. We have, since Ψ is bounded,

(d,r)=1 Ψ Y d 2 = d>D (d,r)=1 Ψ Y d 2 + O(D). (1.43)
Changing the order of summation and integration, we have

d>D (d,r)=1 Ψ Y d 2 = Y D 2 0 ψ(v) D<d≤ Y v (d,r)=1 1dv = Y D 2 0 ψ(v) ϕ(|r|) |r| ((Y /v) 1 2 -D) + O(d(r)) dv = ϕ(|r|) |r| Y 1 2 Y D 2 0 ψ(v)v -1 2 dv + O D + d(r) Y D 2 .
By lemma 1.5.1, the above equation implies Preparatory results

d>D (d,r)=1 Ψ Y d 2 = -2ϕ (|r|) |r| ζ - 1 2 Y 1 2 + O D + d(r) Y D 2 . (1.44)
We make the choice D = Y 1 3 . Formula (1.42) is now just a consequence of (1.43), (1.44) and the functional equation for the Riemann zeta function, which for s = -1/2 gives

ζ - 1 2 = - ζ 3 2 4π
.

We now deduce (1.40) from (1.42). We can write

G(Y, r) = (d 1 ,r)=1 β(d 1 ) (d 2 ,r)=1 Ψ Y d 2 1 d 2 2
.

(1.45)

We have two possibilities. If d 2 1 ≤ Y , we shall use formula (1.42) for the inner sum on the right-hand side of (1.45). If, otherwise, d 2 1 > Y , we shall use the trivial bound

(d 2 ,r)=1 Ψ Y d 2 1 d 2 2 ≤ d 2 Y d 2 1 d 2 2 Y d 2 1
.

Thus, we deduce from (1.45) that

G(Y, r) = ϕ(|r|) |r| ζ( 3 2 ) 2π Y 1 2 d 1 ≤ √ Y (d 1 ,r)=1 β(d 1 ) d 1 + O      d(r)Y 1 3 d 1 ≤ √ Y (d 1 ,r)=1 β(d 1 ) d 2 3 1 + Y d 1 > √ Y (d 1 ,r)=1 β(d 1 ) d 2 1     
.

By completing the first term and using lemma 1.5.2, we have

G(Y, r) = C β (r) ϕ(|r|) |r| ζ( 3 2 ) 2π Y 1 2 + O   d(r)Y 1 3 ab 2 ≤ √ Y d(a) a 8 3 b 4 3   + O   Y 1 2 ab 2 > √ Y d(a) a 3 b 2 + Y ab 2 > √ Y d(a) a 4 b 4   , (1.46)
where

C β (r) = (d 1 ,r)=1 β(d 1 ) d 1 = p 1 - p -2 p(p 2 -2) p|r 1 - p -2 p(p 2 -2) -1
.

The first error term on the right-hand side of (1.46) is clearly d(r)Y 1 3 . For the second one, we have

ab 2 > √ Y d(a) a 3 b 2 ≤ ab>Y 1 4 d(a) a 2 b 2 Y -1 4 (log Y ) 2 ,
1.1.2 -A double sum over squarefree integers 35 and analogously, the last one satisfies

ab 2 > √ Y d(a) a 4 b 4 Y -3 4 (log Y ) 2 .
Once we inject these upper bounds in equation (1.46), we obtain

G(Y, r) = C p|r 1 + p(p 2 -2) -1 -1 Y 1 2 + O d(r)Y 1 3 + Y 1 4 (log Y ) 2 ,
which concludes the proof of formula (1.40).

In the following lemma we gather a series of identities that shall be useful later on, and whose proofs are routine and hence omitted. Lemma 1.5.4. Let m be a squarefree integer and let κ(ρ) be as in (1.26), then we have the equalities

                     ρσ|m 2 κ(ρ)µ(σ) ρσ = p|m p 2 -1 p 2 , ρσ|m 2 κ(ρ)µ(σ) = p|m p 2 -p p 2 -1 , ρσ|m 2 κ(ρ)µ(σ)ρ 1 2 σ 1 2 = p|m p 2 -p 3 2 + p -1 p 2 -1 .
Also, let r = 0 be an integer, and let h(d) be as in (1.38). Then we have further the equalities

             (d,r)=1 h(d) d 4 = p (p 2 -1) 2 p 2 (p 2 -2) p|r (p 2 -1) 2 p 2 (p 2 -2) -1 , (d,r)=1 h(d) d 2 = p p 2 -1 p 2 -2 p|r p 2 -1 p 2 -2 -1
.

Finally, let m = 0 be squarefree and q be a positive integer. Also let f q ( , m) be as in (1.24) and C(q) be as in definition (1.13). Then, we have the following equality

f q (0, m) = ϕ(|mq|) |mq| C(mq). Let S[m](Y, q) := 0< ≤Y f q ( , m)(Y -).
(1.47)

The following Proposition gives a formula for S[m](Y, q) by means of the previous lemmas.

Proposition 1.5.5. Let m = 0 be an integer and let f q ( , m) be defined by (1.24). Then, uniformly for Y, q ≥ 1, we have

S[m](Y, q) = ϕ(q) 2q C(q) 2 Y 2 - ϕ(|mq|) 2|mq| C(mq)Y + C 2 Γ ar (m) p|q (1 + 2p -1 ) -1 Y 1 2 + O m (d(q)Y 1 3 ),
where C and Γ ar (m) are as in (1.7) and (1.11) respectively, C(r) is as in (1.13) for r = q, mq, and the implied O m -constant depends at most on m.

Preparatory results

Démonstration. We recall (1.24) and use Lemma 1.5.4 for the innermost product. We have

f q ( , m) = C 2 p|m p 2 -1 p 2 -2 p|q p 2 -p p 2 -2 κ(( , m 2 )) d 2 | (d,mq) h(d) d 2 ,
where C 2 is as in (1.25). We notice that the first three factors on the right-hand side of the above equation are independent of . Therefore, in order to evaluate S[m](Y, q), we need to study

S [m](Y, q) := 0< ≤Y κ(( , m 2 )) d 2 | (d,mq)=1 h(d) d 2 (Y -) = ρ|m 2 κ(ρ) 0< ≤Y ( ,m 2 )=ρ (Y -) d 2 | (d,mq)=1 h(d) d 2 = ρσ|m 2 κ(ρ)µ(σ)ρσ (d,mq)=1 h(d) d 2 0< 0 ≤ Y ρσ d 2 | 0 Y ρσ -0 (1.48)
where in the third line we used Möbius inversion to detect the coprimality condition. We proceed by writing the inner sum as an integral. We have

0< 0 < Y ρσ d 2 | 0 Y ρσ -0 = 0< 0 ≤ Y ρσ d 2 | 0 Y ρσ 0 1du = Y ρσ 0 0< 0 ≤u d 2 | 0 1du = Y ρσ 0 u d 2 du. (1.49)
Recall formula (1.15) defining the sawtooth function

ψ x = x - 1 2 + ψ(x).
This formula when used in equation (1.49) gives

0< 0 ≤ Y ρσ d 2 | 0 Y ρσ -0 = Y 2 2ρ 2 σ 2 d 2 - Y 2ρσ + d 2 Ψ Y ρσd 2 .
Injecting this formula in (1.48), we deduce the equality

S [m](Y, q) = λ 2 (m, q)Y 2 -λ 1 (m, q)Y + ρσ|m 2 κ(ρ)µ(σ)ρσG Y ρσ , mq , (1.50)
where

λ 2 (m, q) = 1 2 ρσ|m 2 κ(ρ)µ(σ) ρσ × (d,mq)=1 h(d) d 4 , λ 1 (m, q) = 1 2 ρσ|m 2 κ(ρ)µ(σ) × (d,mq)=1 h(d) d 2 ,
and G(Y, q) is as defined in (1.39). The study of the last sum in the right-hand side of (1.50)

is divided in two cases. If Y ρσ ≥ 1, we use formula (1.40) to obtain G Y ρσ , mq = C p|mq 1 + p(p 2 -2) -1 -1 Y ρσ 1 2 + O d(mq) Y ρσ 1 3
.

On the other hand, if Y ρσ < 1, we have both Y ρσ 1 2
1 and

G Y ρσ , mq d≥1 Y ρσd 2 1.
Hence in this case we can write

G Y ρσ , mq = C p|mq 1 + p(p 2 -2) -1 -1 Y ρσ 1 2 + O (1)
.

Combining the estimates above we get

ρσ|m 2 κ(ρ)µ(σ)ρσG Y ρσ , mq = λ(m, q)Y 1 2 + O m (d(q)Y 1 3 ), (1.51)
where

λ(m, q) = C p|mq 1 + p p 2 -2 -1 ρσ|m 2 κ(ρ)µ(σ)ρ 1 2 σ 1 2 .
Now, putting together (1.50) and (1.51), we obtain

S [m](Y, q) = λ 2 (m, q)Y 2 -λ 1 (m, q)Y + λ(m, q)Y 1 2 + O m (d(q)Y 1 3 ). Hence S[m](Y, q) = Λ 2 (m, q)Y 2 -Λ 1 (m, q)Y + Λ(m, q)Y 1 2 + O m (d(q)Y 1 3 ), (1.52)
where

           Λ(m, q) = C 2 p|m p 2 -1 p 2 -2 p|q p 2 -p p 2 -2 λ(m, q), Λ i (m, q) = C 2 p|m p 2 -1 p 2 -2 p|q p 2 -p p 2 -2 λ i (m, q), i = 1, 2.
Lemma 1.5.4 ensures that the constants Λ 2 (m, q), Λ 1 (m, q) and Λ(m, q) correspond to the constants in Proposition (1.5.5). Hence the result follows from (1.52).

Main term

Main term

In this section we show how to use the results from the previous section to exhibit an asymptotic formula for A[m](X, q) (see (1.35)). We prove the following Proposition 1.6.1. For X > 1, and integers m, q such that m is squarefree and q ≥ 1, let A[m](X, q) be defined by formula (1.35). Then we have, uniformly for X, q > 1,

A[m](X, q) = ϕ(q) C(q) X q 2 + C 2 Γ an (m)Γ ar (m) p|q (1 + 2p -1 ) -1 X 1 2 q 1 2 + O m d(q)X 1 3 q 2 3
, where C, C(q), Γ an (m) and Γ ar (m) are as in (1.7), (1.13), (1.10) and (1.11), respectively, and the implied O m -constant depends at most on m.

Démonstration. There is a slight difference depending on whether m > 0 or m < 0. So we study the two cases separately.

The case m > 0

By analyzing the possible values of |I( )|, we have

A[m](X, q) = f q (0, m) X m + 0< ≤ X q f q ( , m) X -q m + - (m-1)X q ≤ <0 f q ( , m) X m + -mX q ≤ <- (m-1)X q f q ( , m) X + q m . (1.53)
We remark that for m, q fixed, f q ( , m) only depends on the positive divisors of (see formula (1.24)). Hence f q ( , m) = f q (-, m). As a consequence, formula (1.53) implies that

A[m](X, q) = f q (0, m) X m + 1 m 0< ≤ X q f q ( , m) (X -q) - 1 m 0< ≤ (m-1)X q f q ( , m) ((m -1)X -q) + 1 m 0< ≤ mX q f q ( , m) (mX -q) .
That is

A[m](X, q) = f q (0, m) X m + q m S[m](X/q, q)-S[m]((m-1)X/q, q)+S[m](mX/q, q) , (1.54)
where S[m](Y, q) is as in (1.47). By Proposition 1.5.5, we have 1.1.2 -A double sum over squarefree integers

S[m](Y, q) = ϕ(q) 2q C(q) 2 Y 2 + ϕ(|mq|) 2|mq| C(mq)Y + C 2 Γ ar (m) p|q (1 + 2p -1 ) -1 Y 1 2 + O m d(q)Y 1 3
.

Hence, (1.54) implies that

A[m](X, q) = f q (0, m) X m + ϕ(q)C(q) 2 X q 2 - ϕ(|mq|) |mq| C(mq) X m + C 2 Γ an (m)Γ ar (m) p|q (1 + 2p -1 ) -1 X 1 2 q 1 2 + O m d(q)X 1 3 q 2 3
, where Γ an (m) is as in (1.10). Now, by lemma 1.5.4, the first and third terms cancel each other out. This concludes the proof of the Proposition when m > 0.

The case m < 0

Analogously to the previous case, we have

A[m](X, q) = q m {S[m](X/q, q) + S[m](-mX/q, q) -S[m]((1 -m)X/q, q)}
and, again by Proposition 1.5.5, the result holds in this case as well.

Bounding S >y [m](X, q)

In the present section we give a bound for S >y [m](X, q) (recall (1.23)). We start by noticing that d 2 | σ(n)σ(mn + q) if and only if there exist j, k such that d = jk and both j 2 | n and k 2 | mn + q. Moreover since we are supposing n, mn + q < X, we have j, k < X 1 2 . From this observation we deduce

S >y [m](X, q) ≤ j,k≤X 1 2 jk>y (jk,q)=1 #{(n, ) ∈ Z 2 ; 0 < n, mn + q < X and j 2 | n, k 2 | mn + q} =: j,k≤X 1 2 jk>y (jk,q)=1 N [m](X, q; j, k), (1.55)
say. We shall divide the possible values of j and k into sets of the form

B(J, K) := {(j, k); (jk, q) = 1, j ∼ J, k ∼ K}.
We can do the division using at most O((log X) 2 ) such sets since we are summing over j, k ≤ X

1 2 . Let N [m](J, K) = (jk,q)=1 j∼J,k∼K N [m](X, q; j, k) = # (j, k, u, v); j ∼ J, k ∼ K, 0 < j 2 u, k 2 v < X, and mj 2 u ≡ k 2 v (mod q) (1.56)
By taking the maximum over all J, K, we obtain a pair (J, K) with J, K ≤ X

1 2 such that S >y [m](X, q) X N [m](J, K). (1.57)
By the condition jk > y in (1.55), we can also impose

JK ≥ y 4 .
Our problem now is to bound N [m](J, K). Notice that, since we bound S >y [m](X, q) as in (1.57), we will not be able to benefit from oscillations of the coefficients µ(d) in (1.23). Although formula (1.56) is not symmetrical with respect to J and K, we would like to benefit from some symmetry. With that in mind, let m 1 , m 2 ∈ Z we define

N [m 1 , m 2 ](J, K) := # (j, k, u, v); j, k ∈ B(J, K), 0 < j 2 u, k 2 v < X,
and

m 1 j 2 u ≡ m 2 k 2 v (mod q) . (1.58) We also suppose max(|m 1 |, |m 2 |) ≤ |m|. (1.59)
In the following we estimate the general N [m 1 , m 2 ](J, K) from which we can directly deduce an estimate for N [m](J, K) itself. The first bound we give is an auxiliary one and will be useful later on.

Auxiliary bound

Lemma 1.7.1. Let X ≥ 1 and let q ≥ 1 be an integer. Also let m, m 1 , m 2 such that 0

< |m 1 |, |m 2 | ≤ |m|. Let N [m 1 , m 2
](J, K) be as in (1.58), then for every J, K ≤ X and every > 0, we have

N [m 1 , m 2 ](J, K) ,m X 2+ q (JK) -1 + J -2 K ,
where the implied constant depends at most on and m.

-Auxiliary bound

Démonstration. Putting q = m 1 j 2 u -m 2 k 2 v, we have that | | ≤ 2|m|X q . Hence we have the inequality

N [m 1 , m 2 ](J, K) ≤ | |≤ 2|m|X q k∼K (k,q)=1 u≤XJ -2 j∼J m 1 j 2 u≡-q (mod k 2 ) 1.
To estimate the above quadruple sum, we make a change of variables. First we write f = (j, k). Since (k, q) = 1, we must have f 2 | . We make a change of variables

j 0 = j f , k 0 = k f and 0 = f 2 .
The congruence in the innermost sum then becomes

m 1 j 2 0 u ≡ -0 q (mod k 2 0 ). (1.60)
Now, let g = (k 2 0 , m 1 ). We have g | 0 . We write

r = k 2 0 g , s = m 1 g and t = 0 g .
Finally, let h = (r, t). This implies that h | u. We write

r = r h , t = t h and u = u h .
So (1.60) becomes su j 2 0 ≡ -t q (mod r ) and since (t q, r ) = 1, it has at most 2 ω(r )+1 ≤ 2d(k) solutions in j 0 (mod t ). Therefore we have

N [m 1 , m 2 ](J, K) ≤ f ≥1 0 ≤ X f 2 q k 0 ∼K/f (k 0 ,q)=1 u ≤XJ -2 h -1 j 0 ∼J/f su j 2 0 ≡-t q (mod r ) 1 ≤ 2 f ≥1 ≤ X f 2 q k 0 ∼K/f XJ -2 h -1 Jgh f k 2 0 + 1 d(k) m f ≥1 ≤ X f 2 q k 0 ∼K/f XJ -2 J f k 2 0 + 1 d(k) f ≥1 X 2+ J 2 f 2 q J K 2 + 1 K X 2+ q 1 JK + K J 2 .
Remark 1.7.1. We certainly have a result similar to Lemma 1.7.1 with the roles of J and and K interchanged.

Square Sieve

We must now proceed to obtain a more precise bound. We start from the equality

N [m 1 , m 2 ](J, K) = u≤XJ -2 N u [m 1 , m 2 ](J, K),
where

N u [m 1 , m 2 ](J, K) = # (j, k, v); j, k ∈ B(J, K), 0 < k 2 v < X and m 1 j 2 u ≡ m 2 k 2 v (mod q) .
In other words,

N u [m 1 , m 2 ](J, K) counts the contribution to N [m 1 , m 2 ](J, K) for a fixed u ≤ XJ -2 .
Again by dyadic decomposition, we see that there is a certain U satisfying

U ≤ XJ -2 , (1.61) such that N [m 1 , m 2 ](J, K) X N [m 1 , m 2 ](J, K, U ), (1.62)
where

N [m 1 , m 2 ](J, K, U ) = u∼U N u [m 1 , m 2 ](J, K).
If we analyze the possible values for v and :

= m 2 k 2 v-m 1 j 2 u q contributing to N u [m 1 , m 2 ](J, K), we obtain N u [m 1 , m 2 ](J, K) ≤ # (j, k, , v); j ∼ J, k ∼ K, m 1 j 2 u = m 2 k 2 v -q, | | ≤ |m|X q , v ≤ XK -2 , (j, k) = 1 ,
for every u ∼ U .

We now appeal to the square sieve as in [START_REF] Heath-Brown | The square sieve and consecutive square-free numbers[END_REF]. We state the main result here for easier reference.

Theorem 1.7.2. ([22, Theorem 1]) Let P be a set of P odd primes and (w(n)) n≥1 a sequence of real numbers. Suppose that w(n) = 0 for n = 0 or n ≥ e P . Then

n≥1 w(n 2 ) P -1 n≥1 w(n) + P -2 p 1 =p 2 p 1 ,p 2 ∈P n w(n) n p 1 p 2 ,
where n p 1 p 2 is the Jacobi symbol and the implied constant is absolute.

1.7.2 -Square Sieve 43 Fix K ≤ X 1 2 , and integers m, m 1 , m 2 such that 0 < |m 1 |, |m 2 | ≤ |m|.
For each u ∼ U , we apply the square-sieve to the multi-set of integers

A u = A u [m, m 1 , m 2 ](K) given by A u = m 1 (m 2 k 2 v -q) u ; u | m 2 k 2 v -q, k ∼ K, ≤ 2|m|X q , v ≤ XK -2 ,
For an integer n, let w u (n) denote the multiplicity of n in A. We have

N u [m 1 , m 2 ](J, K) ≤ j≥1 w u ((m 1 j) 2 ) ≤ n≥1 w u (n 2 ).
For the set of primes, we take

P = p prime ; p 2m 1 m 2 u, P < p ≤ 2 P ,
where P will be chosen later depending on J, K, X, q, and subject to the condition

(log |m| 2 X) 2 ≤ P ≤ |m| 2 X.
(1.63)

We have P = #P ∼ P (log P ) -1 as P → ∞ and thus

w u (n) = 0 for n ≥ e P ,
because, for X sufficiently large,

e P ≥ e P 1 2 ≥ |m| 2 X,
and w u (n) = 0 for n > |m| 2 X. Theorem 1.7.2 and the definition of P then give the inequality

N u [m 1 , m 2 ](J, K) P -1 (log X) n≥1 w u (n) + P -2 p 1 ,p 2 p 1 ,p 2 ∈P p 1 =p 2 k, ,v m 2 k 2 v -q p 1 p 2 , (1.64)
where the conditions in the last sum are

k ∼ K, ≤ Xq -1 , v ≤ XK -2 , u | m 2 k 2 v -q, m 2 k 2 v -q = 0. (1.65)
To simplify, we write

T 1 (u) = P -1 (log X) n≥1 w u (n) T 2 (u) = P -2 p 1 ,p 2 p 1 ,p 2 ∈P k, ,v m 2 k 2 v -q p 1 p 2 .
So that (1.64) implies the inequality

Bounding S >y [m](X, q) N [m 1 , m 2 ](J, K, U ) u∼U T 1 (u) + u∼U T 2 (u).
(1.66)

Study of T 1 (u)
By the definition of w u (n), we have

u∼U T 1 (u) P -1 (log X) k, ,v u|m 2 k 2 v-q 1 P -1 (log X) k, ,v d(m 2 k 2 v -q),
where the conditions on the sum are

k ∼ K, ≤ X q , v ≤ XK -2 , m 2 k 2 v -q ≥ 1.
Using the classical bound for the divisor function, we have

u∼U T 1 (u) K -1 P -1 X 2+ q, (1.67)
for every > 0, where the implied constant depends at most on and m.

Study of T 2 (u)

We have

u∼U T 2 (u) = P -2 p 1 =p 2 u∼U k, ,v m 2 k 2 v -q p 1 p 2 ≤ u∼U max p i 2m 1 m 2 u P <p 1 <p 2 ≤2 P k, ,v m 2 k 2 v -q p 1 p 2 . (1.68) For each u ∼ U , we pick (p 1 , p 2 ) = (p 1 (u), p 2 (u))
for which the maximum is attained, and we proceed to estimate the sum

S u := k, ,v m 2 k 2 v -q p 1 p 2 , (1.69)
where the conditions on k, , v are as in (1.65), and we recall p 1 = p 2 . Thus, (1.68) implies that

u∼U T 2 (u) u∼U |S u | . (1.70)
We write

S u = up 1 p 2 -1 α,β,γ=0 u|m 2 α 2 β-qγ m 2 α 2 β -qγ p 1 p 2 k∼K k≡α (mod up 1 p 2 ) v≤XK -2 v≡β (mod up 1 p 2 ) ≤Xq -1 ≡γ (mod up 1 p 2 ) 1 = α,β,γ u|m 2 α 2 β-qγ m 2 α 2 β -qγ p 1 p 2 1 up 1 p 2 up 1 p 2 λ=1 k∼K e λ(α -k) up 1 p 2 ×    1 up 1 p 2 up 1 p 2 µ=1 v≤XK -2 e µ(β -v) up 1 p 2       1 up 1 p 2 up 1 p 2 -1 ν=0 ≤Xq -1 e ν(γ -) up 1 p 2    =(up 1 p 2 ) -3 up 1 p 2 -1 λ,µ,ν=0 S(u, p 1 p 2 ; m 2 , q; λ, µ, ν)Θ λ Φ µ Ψ ν , (1.71)
where

                                     S(u, p 1 p 2 ; m 2 , q; λ, µ, ν) = up 1 p 2 -1 α,β,γ=0 u|m 2 α 2 β-qγ m 2 α 2 β -qγ p 1 p 2 e λα + µβ + νγ up 1 p 2 , Θ λ = K<k≤2K e -λk up 1 p 2 min K, λ up 1 p 2 -1 , Φ µ = v≤XK -2 e -µv up 1 p 2 min XK -2 , µ up 1 p 2 -1 , Ψ ν = ≤Xq -1 e -ν up 1 p 2 min Xq -1 , ν up 1 p 2 -1 ;
(1.72) here x denotes the distance from x to the nearest integer. The Chinese remainder theorem allows us to write S(u, p 1 p 2 ; m 2 , q; λ, µ, ν) = S 1 (p 1 ; m 2 , q; b, c, d)S 1 (p 2 ; m 2 , q; b, c, d)

r f u S 2 (r f ; m 2 , q; b, c, d),
(1.73) where u = r f is the prime factorization of u, b, c and d are some integers such that

     (b, up 1 p 2 ) = (λ, up 1 p 2 ), (c, up 1 p 2 ) = (µ, up 1 p 2 ), (d, up 1 p 2 ) = (ν, up 1 p 2 ),
and

                 S 1 (p; m 2 , q; b, c, d) = p-1 α,β,γ=0 m 2 α 2 β -qγ p e bα + cβ + dγ p S 2 (r f ; m 2 , q; b, c, d) = r f -1 α,β,γ=0 r f |m 2 α 2 β-qγ e bα + cβ + dγ r f .
For these sums we have the following upper bounds whose proofs are elementary and are given in section 1.9 (see Lemma 1.9.1).

               S 1 (p; m 2 , q; b, c, 0) = 0, S 1 (p; m 2 , q; b, c, d) p 3 2 , |S 2 (r; m 2 , q; b, c, d)| ≤ 2r(r, b, c, dm 2 ) |S 2 (r f ; m 2 , q; b, c, d)| ≤ 2r 3f 2 (r f , b, c, dm 2 ) 1 2 , if r is odd, f ≥ 2, |S 2 (2 f ; m 2 , q; b, c, d)| ≤ 4.2 3f 2 (2 f , b, c, dm 2 ) 1 2 , f ≥ 2.
If we multiply these upper bounds in (1.73), we have, whenever d ≡ 0(mod up 1 p 2 ),

S(u, p 1 p 2 ; m 2 , q; b, c, d) d(u) P 3 U 3 2 (u † ) -1 2 (u, b, c, dm 2 ) m d(u) P 3 U 3 2 (u † ) -1 2 (u, b, c, d), (1.74)
where u † = p|u p 2 u p.

(1.75)

We also have S(u, p 1 p 2 ; m 2 , q; b, c, 0) = 0.

Hence, by (1.71), we deduce the inequality

S u m d(u) P -3 U -3 2 (u † ) -1 2 up 1 p 2 -1 ν=1 up 1 p 2 -1 λ,µ=0 |Θ λ Φ µ Ψ ν |(u, λ, µ, ν). (1.76)
Now, we separate the inner sum in four parts accordingly to whether λ and µ are zero or not. We also use the bounds coming from (1.72). We use the first term inside the min-symbol in the zero case and the second term otherwise. We then have

up 1 p 2 -1 λ,µ=0 up 1 p 2 -1 ν=1 |Θ λ Φ µ Ψ ν |(u, λ, µ, ν) K -1 P 2 U X 1≤ν≤ up 1 p 2 2
ν -1 (u, ν)

+ K P 4 U 2 1≤µ,ν≤ up 1 p 2 2 µ -1 ν -1 (u, µ, ν) + K -2 P 4 U 2 X 1≤λ,ν≤ up 1 p 2 2
λ -1 ν -1 (u, λ, ν)

+ P 6 U 3 1≤λ,µ,ν≤ up 1 p 2 2 λ -1 µ -1 ν -1 (u, λ, µ, ν). (1.77) 1.7.2 -Square Sieve 47
For the sums involving greatest common divisors, we have the following elementary Lemma Lemma 1.7.3. For every > 0, we have the following inequalities

1≤λ≤Z λ -1 (u, λ) (Zu) , (1.78) 1≤λ,µ≤Z λ -1 µ -1 (u, λ, µ) (Zu) , (1.79) 1≤λ,µ,ν≤Z λ -1 µ -1 ν -1 (u, λ, µ, ν) Z , (1.80)
uniformly for Z ≥ 1 and every positive integer u, where the implied constants depend on alone.

If we insert the inequalities (1.78), (1.79) and (1.80) (with Z = up 1 p 2 2 ) in (1.77), we deduce

S u ,m (u † ) -1 2 X K -1 P -1 U -1 2 X + K P U 1 2 + K -2 P U 1 2 X + P 3 U 3 2
.

(1.81)

In order to compute the contribution of S to N [m 1 , m 2 ](J, K, U ) we must sum over u. We notice that u∼U

(u † ) -1 2 ≤ v≤U v -1 2 w∼U/v w squarefull 1 U 1 2 + , (1.82)
by the well-known bound

w≤W w squarefull 1 W 1 2 .
Gathering (1.70), (1.81), and (1.82), we obtain

u∼U T 2 (u) ,m X K -1 P -1 X + K P U + K -2 P U X + P 3 U 2 .
Appealing to (1.61), we deduce the inequality

u∼U T 2 (u) ,m X K -1 P -1 X + J -2 K P X + J -2 K -2 P X 2 + J -4 P 3 X 2 . (1.83)
From (1.66), (1.67) and (1.83), we deduce an inequality for

N [m 1 , m 2 ](J, K, U ) N [m 1 , m 2 ](J, K, U ) ,m X K -1 P -1 X 2 q -1 + K -1 P -1 X + J -2 K P X +J -2 K -2 P X 2 + J -4 P 3 X 2 .
Since q ≤ X, the second term is dominated by the first one. Therefore,

Bounding S >y [m](X, q) N [m 1 , m 2 ](J, K, U ) ,m X K -1 P -1 X 2 q -1 + J -2 K P X +J -2 K -2 P X 2 + J -4 P 3 X 2 .
By taking (m 1 , m 2 ) = (m, 1), we deduce

N [m](J, K, U ) ,m X K -1 P -1 X 2 q -1 + J -2 K P X +J -2 K -2 P X 2 + J -4 P 3 X 2 .
(1.84)

Remark 1.7.2. Notice that since we also have the identity

N [m](J, K) = N [1, m](K, J),
the upper bound (1.84) still holds if we replace the roles of J and K. In other words, there is no loss of generality in supposing J ≥ K.

Now, the upper bound (1.84), together with (1.57) and (1.62) gives us

S >y [m](X, q) ,m X K -1 P -1 X 2 q -1 + J -2 K P X +J -2 K -2 P X 2 + J -4 P 3 X 2 .
(1.85)

At this point we make the choice

P = JK -1 4 q -1 4 + (log |m| 2 X) 2 , (1.86)
which makes the second and the last terms of (1.85) similar. Hence we have

S >y [m](X, q) ,m X J -1 K -3 4 X 2 q -3 4 + J -1 K 3 4 Xq -1 4 + J -1 K -9 4 X 2 q -1 4 + J -2 KX + J -2 K -2 X 2 + J -4 X 2 .
Since J ≥ K, JK y, we see that

S >y [m](X, q) ,m X X 2 q -3 4 y -7 8 + Xq -1 4 y -1 8 + J -1 K -9 4 X 2 q -1 4 + Xy -1 2 + X 2 y -2 .
(1.87)

S[m](X, q) = A[m](X, q) + R[m](X, q), (1.89)
where

R[m](x, q) satisfies R[m](X, q) ,m X Xq -1 y + X 2 q -3 4 y -7 8 + Xq -1 4 y -1 8 + J -1 K -9 4 X 2 q -1 4 + Xy -1 2 + X 2 y -2 ,
thanks to (1.88). We now make the choice

y = X 8 15 q 2 15
(1.90) to make the first two terms similar. Notice that this choice of y satisfies (1.88). Hence we have

R[m](X, q)

,m X X 23 15 q -13 15 + X 14 15 q -4 15

+ J -1 K -9 4 X 2 q -1 4 + X 11 15 q -1 15 X X 23 15 q -13 15 + E 1 , (1.91)
where

E 1 = X 2 J -1 K -9 4 q -1 4
since q ≤ X. Now Lemma 1.7.1 together with (1.22), (1.34), (1.57) and the choice for y give us

R[m](X, q) ,m X Xq -1 y + X 2 J -1 K -1 q -1 + X 2 J -2 Kq -1 X Xq -1 y + X 2 q -1 y -1 + E 2 X X 23 15 q -13 15 + E 2 , (1.92)
thanks to (1.88), and

E 2 = X 2 J -2 Kq -1 .
The importance of Lemma 1.7.1 is that since J ≥ K, we cannot give a good estimate for the term E 1 using that JK y. So we look for a mixed term that lies between E 1 and E 2 for which we have a good bound (much better than the one for E 2 , for example). Notice that

min(E 1 , E 2 ) ≤E 12 17 1 E 5 17 2 =X 2 (JK) -22 17 q -8 17 X 2 q -

Exponential Sums

Here we give the proofs for the bounds on exponential sums used in section 1.7. We have the following Lemma 1.9.1. Let b, c, d, m 2 , p and q be integers, m 2 = 0, p a prime number and p m 2 q. We define

S 1 (p; m 2 , q; b, c, d) = p-1 α,β,γ=0 m 2 α 2 β -qγ p e bα + cβ + dγ p .
Let r be a prime number such that r q and f > 0 an integer,

S 2 (r f ; m 2 , q; b, c, d) = r f -1 α,β,γ=0 r f |m 2 α 2 β-qγ e bα + cβ + dγ r f .
Then, we have That proves (1.97) when r | dm 2 .

                 S 1 (p; m 2 , q; b, c, 0) = 0, (1.95) S 1 (p; m 2 , q; b, c, d) p 3 2 , (1.96) |S 2 (r; m 2 , q; b, c, d)| ≤ 2r(r, b, c, dm 2 ), (1.97) |S 2 (r f ; m 2 , q; b, c, d)| ≤ 2r 3f 2 (r f , b, c, dm 2 ) 1 2 if r is odd, f ≥ 2, (1.98) |S 2 (2 f ; m 2 , q; b, c, d)| ≤ 4.2 3f 2 (2 f , b, c, dm 2 ) 1 2 , f ≥ 2. ( 1
We now assume r dm 2 . We analyze when the symbol δ(c + dm 2 qα 2 ) is non-zero. That means we consider the equation

dm 2 α 2 ≡ -cq (mod r).
This equation has at most two solutions for 1 ≤ α ≤ r. Thus, again by equation (1.100),

|S 2 (r; m 2 , q; b, c, d)| = r-1 α=0 e bα r δ(c + dm 2 qα 2 , r) ≤ 2r,
which completes the proof of (1.97).

We proceed to prove the inequalities (1.98) and (1.99). Analogously to the previous case, we have

S 2 (r f ; m 2 , q; b, c, d) = r f -1 α=0 e bα r f δ(c + dm 2 qα 2 , r f ).

Exponential Sums

We write (c, r f ) = r s , (dm 2 , r f ) = r t , So we have 0 ≤ s, t ≤ f . If s < t, for any α, the largest power of r that divides c + dm 2 qα 2 is always r s . So the δ symbol is always zero. Hence the sum is itself always zero.

Hence we may suppose s ≥ t. In this case, we write c = r s c, dm 2 = r t d. Then, the condition

r f | c + dm 2 qα 2 is equivalent to r f -t | r s-t c + dqα 2 .
(1.101)

Notice that for any α for which (1.101) is true, we must have

r u | α,
where u = s-t 2 . We write α = r u α, with 1 ≤ α ≤ r f -u . Condition (1.101) now translates to

r f -s | c + r 2u-s+t dq α2 , (1.102)
which has at most 2 solutions for α (mod r f -s ), if r is odd.

Remark 1.9.1. If r = 2 the quadratic equation above can have up to 4 solutions (mod r f -s ), which is in fact the only difference between the cases r odd and r = 2.

In the following, we only prove (1.98). The proof of (1.99) is exactly the same, except that one must take into account Remark 1.9.1. We now have

S 2 (r f ; m 2 , q; b, c, d) = r f r f -u -1 α=0 e bα r f -u = r f r f -s -1 α=0 e bα r f -u r s-u h=0 e br f -s h r f -u = r f δ(b, r s-u ) r f -s -1 α=0 e bα r f -u , (1.103)
where the in the sum means that we only sum over the α satisfying (1.102). From (1.103), we see that S 2 (r f ; m 2 , q; b, c, d) is zero unless

r s-u | b. (1.104)
Hence we may assume (1.104). Then, (1.103) gives the inequality

|S 2 (r f ; m 2 , q; b, c, d)| ≤ 2r f +s-u . (1.105)
And since u ≥ s-t 2 , we have

f + s -u ≤ f + s + t 2 ≤ 3f 2 + t 2 .
As a consequence, (1.105) becomes

|S 2 (r f ; m 2 , q; b, c, d)| ≤ 2r 3f 2 (r t ) 1 2 .
(1.106)

We want to prove that

(r f , b, c, dm 2 ) = r t . ( 1
S 1 (p; m 2 , q; b, c, d) = p-1 α,β=0 p-1 h=1 h p e bα + cβ + dq(m 2 α 2 β -h) p = p-1 h=1 h p e -dqh p S 2 (p; m 2 , q; b, c, d).
Hence, we see that (1.96) follows from (1.97) and the well-known estimates for Gauss sums. Finally, we see that (1.95) follows from the equation above and the identity

p-1 h=1 h p = 0.
This concludes the proof of the lemma.

Chapitre 2

On Bourgain's Bound and applications to squarefree numbers

Introduction

As usual, let e(x) := e 2iπx , for x ∈ R.

In a recent paper, Bourgain [4] proved a non trivial bound for exponential sums such as n≤N (n,q)=1 e an 2 q ,

where q > 1 is an integer and n denotes the multiplicative inverse of n (mod q). His result holds in the range N ≥ q , for an arbitrarily small, but fixed, > 0. In his paper, Bourgain was interested in an application related to the size of fundamental solutions D > 1 to the Pell equation t 2 -Du 2 = 1.

He followed the lead of Fouvry [START_REF] Fouvry | On the size of the fundamental solution of the Pell equation[END_REF], who suggested that such an upper bound could help to improve the lower bounds for the counting function

S f (x, α) := # ( D , D); 2 ≤ D ≤ x, D is not a square, and D ≤ D 1 2 +α ,
for small values of α. We are interested in a different application of Bourgain's result (see Proposition 2.4.2 below) related to squarefree numbers in arithmetic progressions.

Let X ≥ 1. Let a and q be coprime integers such that q ≥ 2. We let

E(X, q, a) := n≤X n≡a (mod q) µ 2 (n) - 6 π 2 p|q 1 - 1 q 2 -1 X q .
(2.1)

For fixed q, the last term is known to be asymptotically equivalent to

1 ϕ(q) n≤X (n,q)=1 µ 2 (n)
as X → ∞. So that E(X; q, a) can be seen as an error term of the distribution of squarefree numbers in arithmetic progressions. One naturally has the trivial bound

E(X, q, a) ≤ X q + 1. (2.2)
In Chapter 1, we proved Theorem 2.1.1. There exists an absolute constant C > 0, such that, for every > 0, we have

a (mod q) (a,q)=1 E(X, q, a) 2 ∼ C p|q 1 + 2p -1 -1 X 1/2 q 1/2 , (2.3)
for X → ∞, uniformly for integers q satisfying X 31/41+ ≤ q ≤ X 1-.

This theorem gives the asymptotic variance of the above mentioned distribution. Inspired by an equivalent problem considered by Fouvry et al [9, Theorem 1.5.], we studied how E(X, q, a) correlates with E(X, q, γ(a)) for suitable choices of γ : Z/qZ → Z/qZ. It is natural to choose γ to be an affine linear map, i.e. γ r,s (a) = ra + s,

where r, s ∈ Z, r = 0 are fixed. Thus our object of study is the following correlation sum

C[γ r,s ](X, q) := a (mod q) a =0,γ -1 r,s (0)
E(X, q, a)E(X, q, γ r,s (a)),

for q prime. In Chapter 1, we already considered the case s = 0 and we found that correlation always existed for any non zero value of r. In particular, there exists C r = 0 such that for X → ∞, X 31/41+ ≤ q ≤ X 1-, one has

C[γ r,0 ](X, q) ∼ C r     a (mod q) (a,q)=1 E(X, q, a) 2     .
(2.6)

Our main result is the following theorem which exhibits a certain independence between the functions a → E(X, q, a) and a → E(X, q, γ r,s (a)) considered as random variables on Z/qZ, which confirms our intuition on this question when γ r,s is not a homothety.

Theorem 2.1.2. There exists an absolute δ > 0 such that for every > 0 and for every integer r = 0, there exists C ,r such that one has the inequality

56 Notation C[γ r,s ](X, q) ≤ C ,r q 1+ + X 1/2 q 1/2 (log q) -δ + X 5/3+ q + X q 2 (2.7)
uniformly for X ≥ 2, integers s and prime numbers q ≤ X such that q rs.

A consequence of Theorems 2.1.1 and 2.1.2 (not necessarily with the same ) is the following Corollary 2.1.3. For every > 0 and r = 0, there exists a function Φ ,r : R + → R + , tending to zero at infinity, such that for every X ≥ 2, for every integer s and for every prime q such that q rs and X 7/9+ ≤ q ≤ X 1-, one has the inequality

C[γ r,s ](X, q) ≤ Φ ,r (X)     a (mod q) (a,q)=1 E(X, q, a) 2     .
(2.8) Inequality (2.8) shows a behavior different from (2.6) corresponding to the case where q | s. In other words, it suggests a certain independence of the random variables a → E(X, q, a) and a → E(X, q, γ r,s (a))

Here, as in Chapter 1, we give results that are true for a general r = 0, but in order to simplify the presentation, we give proofs that are only complete when r is squarefree (the case where µ 2 (r) = 0 implies a more difficult definition of the κ function in (2.27)).

Notation

We define the Bernoulli polynomials B k (x) for k ≥ 1, on [0, 1), in the following recursive way

B 1 (x) := x -1/2 d dx B k+1 (x) = B k (x), 1 0 B k (x)dx = 0.
We can extend these functions to periodic functions defined in the whole real line by posing

B k (x) := B k ({x}).
We further notice that B 1 (x) satisfies the following relation

x = x - 1 2 -B 1 (x) (2.9)
and B 2 (x) satisfies

B 2 (x) = x 2 2 - x 2 + 1 12 for 0 ≤ x ≤ 1.
(2.10)

ON BOURGAIN'S BOUND AND APPLICATIONS TO SQUAREFREE NUMBERS 57

In the course of the proof oh Theorem 2.1.2 we will make repetitive use of the following multiplicative function

h(d) = µ 2 (d) p|d 1 -2p -2 -1 . (2.11)
We also define here the closely related product

C 2 = p 1 - 2 p 2 .
(2.12)

We denote, as usual, by d(n) and d 3 (n) the classical binary and ternary divisor functions, respectively. We write ω(n) for the number of primes dividing n. We write n ∼ N as an alternative to N < n ≤ 2N . If S is a finite set, #S denotes its cardinality. If I ⊂ R is an interval, |I| denotes its length. We use indistinguishably the notations f = O(g) and f g when there is an absolute constant C such that |f | ≤ Cg, on a certain domain of the variables which will be clear by the context, and the the same for the symbols O , O r , O ,r and , r , ,r , but with constants that may depend on the subindexed variables.

Initial Steps

Let X ≥ 2. Let γ = γ r,s be given by (2.4) and let q be a prime number ≤ X such that q rs. We start by completing the sum defining C[γ](X, q) (see (2.5)) and we bound trivially the additional terms. By (2.2), we see that

C[γ](X, q) = q-1 a=0 E(X, q, a)E(X, q, γ(a)) + O X q 2 , (2.13)
In what follows, for simplification, we shall write

C(q) = 6 π 2 1 - 1 q 2 -1
.

(2.14)

As we develop the sum on the right-hand side of (2.13), we obtain

C[γ](X, q) = S[γ](X, q) -2C(q) X q n≤X µ 2 (n) + C(q) 2 X 2 q + O X 2 q 2 , (2.15)
where S[γ](X, q) is defined by the double sum

S[γ](X, q) = n 1 ,n 2 ≤X n 2 ≡γ(n 1 ) (mod q) µ 2 (n 1 )µ 2 (n 2 ).
(2.16)

We point out that S[γ](X, q) is the only difficult term appearing in equation (2.15), since we have the well-known formula

n≤X µ 2 (n) = 6 π 2 X + O( √ X) = C(q)X + O X q 2 + √ X , (2.17)
uniformly for 1 ≤ q ≤ X. An asymptotic expansion of S[γ](X, q) will be given in Proposition 2.5.1.

Useful lemmata

We start with a lemma concerning the multiplicative function h(d) which will follow easily from Lemma 1.5.2. Then

β(m) satisfies m≥M β(m) m M -1 2 , (2.18) m≤M β(m) M, (2.19)
uniformly for every M ≥ 1.

Démonstration. By Lemma 1.5.2, we know that β(m) is supported on cubefree numbers and, if we write m = ab 2 with a, b squarefree and relatively prime, then

β(m) d(a) a 2 .
In particular, β(m) 1, which is sufficient to prove (2.19). In order to prove (2.18), we notice that

m≥M β(m) m ab 2 ≥M d(a) a 3 b 2 M -1 2 ∞ a=1 d(a) a 2 M -1 2 .
The next proposition is the main result from [4], which is crucial to our proof.

Proposition 2.4.2. (see [4, Proposition 4]

) There exist constants c, C, C such that for every N, q ≥ 2 and 1 log 2N < β < 1 10 , there exist a subset E N ⊂ {1, 2, . . . , N }(independent of q) satisfying

|E N | ≤ C β log 1 β C N (2.20)
and such that, uniformly for (a, q) = 1, one has

n≤N n ∈E N ,(n,q)=1 e an 2 q ≤ C (log 2N ) C N 1-c β log N log q C . (2.21)
In fact we need the following corollary.

Corollary 2.4.3. There exists an absolute δ > 0 such that for every > 0, we have n≤N (n,q)=1 e an 2 q N (log q) -δ , uniformly for N, q ≥ 2 and N ≥ q .

Remark 2.4.1. More generally, we may consider the sum

Σ(I, q) = n∈I (n,q)=1 e an 2 q
where I is a general interval of length N (mod q). By the completion of exponential sums and Weil's bound for complete sums, we know that

Σ(I, q) q 1/2 log q, (2.22)
for prime numbers q. Hence, (2.22) is non trivial as soon as N ≥ q 1 2 + (for any > 0). Obvioulsy, Bourgain's result is much stronger than (2.22), but it only applies to intervals starting at 1.

Démonstration. (of Corollary 2.4.3)

We use Proposition 2.4.2 and make the choice β = (log N ) -δ 1 , where δ 1 = min 1 2 , 1 2C . We add together inequalities (2.20) and (2.21) to obtain n≤N,(n,q)=1 e an 2 q N (log log

N ) C (log N ) -δ 1 + N (log N ) C exp(c C (log N ) 1/2 )
.

The corollary now follows by taking, for example, δ = δ 1 /2.

Remark 2.4.2. Corollary 2.4.3 will be essential to the proof of Proposition 2.5.1, in which we use it for values of N which are roughly of size X q . Since we want to take q as large as X 1-, it is very important that Bourgain's result holds for N as small as q .

The next lemma is very similar in essence to many others to be found in literature, for example [37, (2.24)

Then, for every r > 0, we have the equality

S( , r) = f ( , r)|I(X, , r)| + O r d 3 ( )X 2/3 (log 2X) 7/3 , (2.25)
uniformly for X ≥ 2 and integers , where

f ( , r) = C 2 p|r p 2 -1 p 2 -2 p 2 | p r p 2 -1 p 2 -2 κ(( , r 2 )), (2.26)
where κ is the multiplicative function defined by

κ(p α ) =            p 2 -p -1 p 2 -1 , if α = 1, p 2 -p p 2 -1 , if α = 2, 0, if α ≥ 3.
(2.27)

We recall that C 2 is defined in (2.12).

Démonstration. We start by defining

σ(n) = p 2 |n p, n = 0, and ξ(n) = σ(n)σ(rn + ).
(2.28)

Notice that the right-hand side of equation (2.28) above actually depends on and r, but since these numbers will be held fixed in the following calculations, we omit this dependency. Since ξ(n) is an integer ≥ 1 and since

µ 2 (n)µ 2 (rn + ) = 1 ⇐⇒ ξ(n) = 1,
we deduce the equality S( , r) = Notice that the condition p | ξ(n)

only depends on the congruence class of n (mod p 2 ), for fixed values of and r. We let

u p ( , r) := # 0 ≤ v ≤ p 2 -1; ξ(v) ≡ 0 (mod p) , (2.29)
and

U d ( , r) := p|d u p ( , r).
Then, by the Chinese remainder theorem, we have the equality

N d ( , r) = U d ( , r) |I(X, , r)| d 2 + O (U d ( , r)) , (2.30)
for every positive squarefree integer d. We also notice that if (p, r) = 1, then |u p ( , r)| ≤ 2 and that |u p ( , r)| ≤ p 2 in general. Therefore we have the upper bound

U d ( , r) r 2 ω(d) .
Let 2 ≤ y ≤ X be a parameter, which will be chosen later to be a power of X. We multiply formula (2.30) by µ(d) and sum for d ≤ y, thus obtaining the equality

d≤y µ(d)N d ( , r) = d≤y µ(d)U d ( , r) |I(X, , r)| d 2 + O r   d≤y 2 ω(d)   . (2.31)
By completing the first sum on the right-hand side of (2.31), we have from which we will deduce the result.

We notice that d | ξ(n) if and only if there exist j, k ≥ 1 such that d = jk, j 2 | n and k 2 | rn + . Moreover since n, rn + < X, we have j, k < √ X. From this observation we deduce

|N >y ( , r)| = y<d≤X µ(d) n ∈ I(X, , r); ξ(n) ≡ 0 (mod d) ≤ j,k≤ √ X jk>y n ∈ Z; 0 < n, rn + < X and j 2 | n, k 2 | rn + = j,k≤ √ X jk>y N (j, k),

say.

We shall divide the possible values of j and k into sets of the form B(J, K) := (j, k) ∈ Z 2 ; j ∼ J, k ∼ K .

We can do the division using at most O((log X) 2 ) of these sets since we are summing over j, k ≤ X 1/2 . Let

N (J, K) := j∼J,k∼K N (j, k) = # (j, k, u, v); j ∼ J, k ∼ K, 0 < j 2 u, k 2 v < X, and k 2 v = rj 2 u +
By taking the maximum over all J, K, we obtain a pair (J, K) with J, K ≤ X 1/2 such that JK ≥ y/4 and we have the upper bound

N >y ( , r) N (J, K)(log X) 2 . (2.33)
At last, we estimate N (J, K) in the following way :

N (J, K) ≤ k∼K u≤XJ -2 j∼J j 2 ru≡-(mod k 2 ) 1.
For j, k relevant to the sum above, we write f = (j, k). From the congruence condition in the inner sum, we have that f 2 | . So we write

j 0 = j f , k 0 = k f and 0 = f 2 .
The congruence then becomes j 2 0 ru ≡ -0 (mod k 2 0 ). Now, let g = (k 2 0 , r) as above we have g | 0 . We write

k 1 = k 2 0 g , s = r g and t = 0 g .
That transforms the congruence into j 2 0 su ≡ -t (mod k 1 ).

Finally, let h = (k 1 , t). From the considerations above, we must have h | u. We write

k = k 1 h , t = t h and u = u h .
So the congruence becomes j 2 0 su ≡ -t (mod k ) and since (t , k ) = 1, it has at most 2.2 ω(k ) ≤ 2d(k 0 ) solutions in j 0 (mod k ). Therefore we have

N (J, K) ≤ g|r f 2 h| k 0 ∼K/f gh|k 2 0 u ≤XJ -2 h -1 j 0 ∼J/f j 2 0 su ≡-t (mod k 2 0 /gh) 1 ≤ 2 g|r f 2 h| k 0 ∼K/f XJ -2 h -1 Jgh f k 2 0 + 1 d(k 0 ) r f 2 h| k 0 ∼K/f XJ -2 J f k 2 0 + 1 d(k 0 ) f 2 h| XJ -2 J K 2 + 1 f K log K d 3 ( )XJ -2 J K 2 + 1 K log X. Hence N (J, K) r d 3 ( ) Xy -1 + XJ -2 K log X.
A similar inequality with the roles of J and K interchanged on the right hand side can be obtained in an analogous way. Combining the two formulas, we deduce

N (J, K) r d 3 ( ) Xy -1 + X(JK) -1/2 log X d 3 ( )Xy -1/2 log X. ( 2
S( , r) = p 1 - u p ( , r) p 2 |I(X, , r)| + O r y log y + d 3 ( )Xy -1/2 (log X) 3 .
We make the choice y = X 2/3 (log X) 4/3 obtaining

S( , r) = p 1 - u p ( , r) p 2 |I(X, , r)| + O r d 3 ( )X 2/3 (log X) 7/3 . (2.35)
We finish by a study of u p ( , r). We distinguish five different cases (we recall that r is squarefree)

-If p | r, p 2 | then u p ( , r) = p, -If p | r, p | but p 2 then u p ( , r) = p + 1, -If p | r, p then u p ( , r) = 1, Useful lemmata -If p r, p 2 | then u p ( , r) = 1, -If p r, p 2 then u p ( , r) = 2.
The lemma is now a consequence of formula (2.35) and the different values of u p ( , r).

Sums involving the B 2 function

In the following we study certain sums involving the Bernoulli polynomials B 2 (x). In the next lemma, we deal with the simplest case

A(Y ; q, a) = n≥1 (n,q)=1 B 2 Y 2 n 2 + an 2 q -B 2 an 2 q , (2.36)
where Y is a positive real number, a, q are coprime integers. The sum above will serve as an archetype for more complicated sums appearing in the proof of Proposition 2.4.6, which in turn will be central for estimating C[γ](X, q). One elementary bound for A(Y ; q, a) can be given by noticing that we have both

B 2 Y 2 n 2 + an 2 q -B 2 an 2 q 1, (2.37)
since B 2 is bounded, and

B 2 Y 2 n 2 + an 2 q -B 2 an 2 q = Y 2 n 2 + an 2 q an 2 q B 1 (v)dv Y 2 n 2 , (2.38)
since B 1 is also a bounded function. Gathering (2.37) and (2.38), we obtain

A(Y ; q, a) n≤Y 1 + n>Y Y 2 n 2
Y.

(2.39)

In the following lemma we give a non-trivial bound for the sum above by means of Bourgain's bound, via Corollary 2.4.3. What we obtain is better than trivial by just a small power of log q, but it is sufficient to obtain Theorem 2.1.2.

Lemma 2.4.5. There exists δ > 0 such that for every > 0, we have the inequality

A(Y ; q, a) Y (log q) -δ , (2.40)
uniformly for integers a and q such that q ≥ 2, (a, q) = 1 and real numbers Y > q .

2.4.1 -Sums involving the B 2 function 65 Démonstration. By Corollary 2.4.3, we know that there exists δ 1 > 0 such that n≤Y (n,q)=1 e an 2 q Y (log q) -δ 1 , (2.41) uniformly for (a, q) = 1 and Y > q /10 . For simplificity, we write

∆ Y (n; q, a) = B 2 Y 2 n 2 + an 2 q -B 2 an 2 q .
(2.42)

The sum on the left-hand side of (2.41) appears naturally once we use the Fourier series development for B 2 (x)

B 2 (x) = h =0 1 4π 2 h 2 e(hx)
(2.43) in formula (2.36). Let θ(q) = (log q) δ 1 /2 .

(2.44) By (2.37) and the Fourier decomposition of B 2 (x) (2.43), we have

n≤Y θ(q) (n,q)=1 ∆ Y (n; q, a) = Y θ(q) -1 ≤n≤Y θ(q) (n,q)=1 ∆ Y (n; q, a) + O(Y θ(q) -1 ) = h =0 1 4π 2 h 2 Y θ(q) -1 ≤n≤Y θ(q) (n,q)=1 e hY 2 n 2 -1 e ahn 2 q + O(Y θ(q) -1 ) = 1≤|h|≤θ(q) 3 1 4π 2 h 2 Y θ(q) -1 ≤n≤Y θ(q) (n,q)=1 e hY 2 n 2 -1 e ahn 2 q + O(Y θ(q) -1 ). (2.45)
Summing by parts, we see that the inner sum of the right-hand side of inequality (2.45) is

Y θ(q) -1 ≤m≤Y θ(q) |h|Y 2 m 3 Y θ(q) -1 ≤n≤m (n,q)=1 e ahn 2 q + Y θ(q) -1 ≤n≤Y θ(q) (n,q)=1 e ahn 2 q .
Now, if q is prime and sufficiently large, then any integer h satisfying 1 ≤ |h| ≤ θ(q) 3 is coprime with q. Then, by (2.41), the above expression is

Y θ(q) -1 ≤m≤Y θ(q) |h|Y 2 m 2 (log q) -δ 1 + Y θ(q) -1
|h|Y θ(q) -1 .

(2.46)

As we insert the upper-bound (2.46) in formula (2.45), we obtain n≤Y θ(q) (n,q)=1 ∆ Y (n; q, a) Y θ(q) -1 log log q Y (log q) -δ 1 /4 .

(2.47)

For the remainder terms we use the trivial upper bound (2.38) to deduce the inequality

n>Y θ(q) (n,q)=1 ∆ Y (n; q, a) n>Y θ(q) Y 2 n 2 Y θ(q) -1 .
(2.48)

Combining (2.47) and (2.48), we obtain that

n≥1 (n,q)=1 ∆ Y (n; q, a) Y (log q) -δ 1 /4 ,
uniformly for (a, q) = 1 and Y > q . The proof of lemma 2.4.5 is now complete.

Remark 2.4.3. Among the hypotheses of lemma (2.4.5), it is essential that we have (a, q) = 1.

In the case where q | a, one cannot improve on (2.39). Indeed, it is possible to show that (see formula (1.42))

A(Y ; q, 0) = - ϕ(q) q ζ(3/2) 2π Y + O(d(q)Y 2/3 ) (Y ≥ 1).

A consequence of Lemma 2.4.5

In order to evaluate S[γ](X, q) (see (2.16)), it is important to consider the following sum which appears in equation (2.25).

Definition 2.4.1. For integers q, r, s such that q ≥ 1 and q rs, let

S[γ r,s ](X, q) := ≡s (mod q) f ( , r)|I(X, , r)|.
Remark that the sum defined above is actually a finite sum since whenever | | > 2|r|X, we have I(X, , r) = ∅. The purpose of this subsection is to prove the following. Proposition 2.4.6. Let C(q) be as in (2.14). There exists δ > 0 and every > 0, for every r = 0 such that r is squarefree, one has

S[γ r,s ](X, q) = 6 π 2 2 1 + 1 q 2 (q 2 -2) -1
X 2 /q + O ,r (q 1+ + X 1/2 q 1/2 (log q) -δ ), (2.49) uniformly for X ≥ 2, for integers s and prime numbers q psuch that q rs.

-A consequence of Lemma 2.4.5 67

The special case r = 1 simplifies many of the calculations in the proof below. For instance, the sums over ρ, σ and τ disappear. Although, this simpler result is, in fact, equally deep and it shows more clearly the connection between the upper bound (2.40) and the error term in (2.49).

Démonstration. We start by recalling (2.26)

f ( , r) = C 2 p|r p 2 -1 p 2 -2 p 2 | p r p 2 -1 p 2 -2 κ(( , r 2 )),
where C 2 is as in (2.12). We notice that the term

C 2 p|r p 2 -1 p 2 -2
is independent of . We consider the sum

S [γ r,s ](X, q) = C -1 2 p|r p 2 -1 p 2 -2 -1 S[γ r,s ](X, q) (2.50) = ≡s (mod q) |I(X, , r)| p 2 | p r p 2 -1 p 2 -2 κ(( , r 2 )).
We expand the product

p 2 | p r p 2 -1 p 2 -2 as follows. p 2 | p r p 2 -1 p 2 -2 = d 2 | (d,r)=1 h(d) d 2 ,
from which we deduce

S [γ r,s ](X, q) := ρ|r 2 κ(ρ) ≡s (mod q) ( ,r 2 )=ρ |I(X, , r)| d 2 | (d,r)=1 h(d) d 2 = ρσ|r 2 κ(ρ)µ(σ) 0 ≡ρσs (mod q) |I (X, ρσ 0 , r) | d 2 | 0 (d,r)=1 h(d) d 2 = ρσ|r 2 κ(ρ)µ(σ) (d,qr)=1 h(d) d 2 1 ≡(ρσd 2 )s (mod q) I(X, ρσd 2 1 , r) (2.51)
where in the second line we used Möbius inversion formula for detecting the gcd condition and we noticed that the congruence satisfied by 0 implies (d, q) = 1.

Useful lemmata

We write the inner sum as an integral :

1 ≡(ρσd 2 )s (mod q) I(X, ρσd 2 1 , r) = X 0 1 ≡(ρσd 2 )s (mod q) 1 (0,X) (ru + ρσd 2 1 )du, (2.52)
where 1 (0,X) is the characteristic function of the interval (0, X). Hence the inner sum above equals

X -ru ρσd 2 q - (ρσd 2 )s q - -ru ρσd 2 q - (ρσd 2 )s q = X ρσd 2 q -B 1 X -ru ρσd 2 q - (ρσd 2 )s q + B 1 -ru ρσd 2 q - (ρσd 2 )s q ,
for almost all u ∈ (0, X) in the sense of Lebesgue measure. If we use this formula in equation (2.52), we deduce the equality

1 ≡(ρσd 2 )s (mod q) I(X, ρσd 2 1 , r) = X 2 ρσd 2 q - ρσd 2 q r B 2 X 2 ρσd 2 q - (ρσd 2 )s q -B 2 - (ρσd 2 )s q -B 2 (1 -r)X ρσd 2 q - (ρσd 2 )s q + B 2 -rX ρσd 2 q - (ρσd 2 )s q . (2.53)
From this point on, we suppose r < 0. The case r > 0 requires only minor modifications. With this hypothesis, we have that both

(1 -r)X ρσd 2 q and -rX ρσd 2 q are positive for every ρ, σ ≥ 1.

We inject (2.53) above in equation ( 2.51) and we define B(D; q, a; r) :=

(d,qr)=1 h(d)∆ D (d, q; a),
where ∆ D (d, q; a) is as in (2.42). From (2.51) and (2.53) we deduce the equality

S [γ r,s ](X, q) = λ(q, r) X 2 q - q r G X q ; q, -s; r -G (1 -r)X q ; q, -s; r + G -rX q ; q, -s; r , (2.54)
where G(Y ; q, s; r) = ρσ|r 2 κ(ρ)µ(σ)ρσB Y ρσ , q, ρσs; r , and λ(q, r) =

ρσ|r 2 κ(ρ)µ(σ) ρσ × (d,qr)=1 h(d) d 4 .
Returning to the function β(m) defined in Lemma 2.4.1, we observe that for a general D > 0, one has

B (D; q, a; r) = (m,qr)=1 β(m) (n,qr)=1 ∆ D (mn; q, a) = (m,qr)=1 β(m) (n,qr)=1 ∆ D/m (n; q, m 2 a) = (m,qr)=1 β(m) τ |r µ(τ ) (n,q)=1 ∆ D/τ m (n; q, τ 2 m 2 a) = (m,qr)=1 β(m) τ |r µ(τ )A(D/τ m, q; τ 2 m 2 a).
We apply the equality above with D = Y ρσ and a = ρσs, multiply by κ(ρ)µ(σ)ρσ and sum over ρ, σ such that ρσ | r 2 , we have

G(Y ; q, s; r) = ρσ|r 2 τ |r (m,qr)=1 κ(ρ)µ(σ)µ(τ )ρσβ(m)A Y ρστ 2 m 2 ; q, ρστ 2 m 2 s . (2.55)
Our discussion depends on the size of Y .

-If Y ≤ q , we have the trivial bound (see (2.39))

A Y ρστ 2 m 2 ; q, ρστ 2 m 2 s Y ρστ 2 m 2 ≤ Y 1/2 m ,
for every ρ, σ, τ ≥ 1. Summing over ρ, σ, τ and m, it gives

G(Y ; q, s; r) r Y 1/2 m≥1 β(m) m q /2 , (2.56)
as a consequence of upper bound (2.18).

-If Y > q , we separate the quadruple sum on the right-hand side of (2.55) as Useful lemmata

m≤q /2 ρστ 2 m 2 >Y /q + m≤q /2 ρστ 2 m 2 ≤Y /q + m>q /2 .
For the first sum we have, again, the trivial bound

A Y ρστ 2 m 2 ; q, ρστ 2 m 2 s Y ρστ 2 m 2 ≤ q /2 ,
(2.57)

The most delicate sum is the second one, since we appeal to (2.40). This gives

A Y ρστ 2 m 2 ; q, ρστ 2 m 2 s Y ρστ 2 m 2 (log q) -δ .
(2.58)

For the third one, we use the trivial bound,

A Y ρστ 2 m 2 ; q, ρστ 2 m 2 s Y ρστ 2 m 2 , ( 2
G(Y ; q, s; r) ,r q /2 m≤q /2 |β(m)| + √ Y (log q) -δ m≤q /2 |β(m)| m + √ Y m>q /2 |β(m)| m ,
and finally, by Lemma 2.4.1

G(Y ; q, s; r) ,r q + √ Y (log q) -δ (Y > q ).
(2.60)

Comparing with (2.56), we have that (2.60) is true for any Y ≥ 1.

Combining (2.60) and (2.54), one has S [γ r,s ](X, q) = λ(q, r) X 2 q + O ,r (q 1+ + X 1/2 q 1/2 (log q) -δ ).

(2.61)

If we multiply the formula above by

C 2 p|r p 2 -1 p 2 -2 (recall formula (2.50)), we deduce S[γ r,s ](X, q) = Λ(q, r) X 2 q + O ,r (q 1+ + X 1/2 q 1/2 (log q) -δ ), (2.62)
where

Λ(q, r) = C 2 p|r p 2 -1 p 2 -2 ρσ|r 2 κ(ρ)µ(σ) ρσ × (d,qr)=1 h(d) d 4
Since for r squarefree, we have the equality

ρσ|r 2 κ(ρ)µ(σ) ρσ = p|r p 2 -1 p 2 ,
then, by some standard calculations, we notice that Λ(q, r) does not depend on r. More precisely, since, q is prime and (q, r) = 1, we have

Λ(q, r) = 6 π 2 2 1 + 1 q 2 (q 2 -2) -1 .
As a consequence, formula (2.62) completes the proof of Proposition 2.4.6.

Study of S[γ r,s ](X, q)

We rewrite S[γ r,s ](X, q) (see (2.16)) as

S[γ r,s ](X, q) = ≡s(mod q) n∈I(X, ,r) µ 2 (n)µ 2 (rn + ).
(2.63)

Recall that for | | > 2|r|X we have I(X, , r). Hence, by (2.25), we have that (recall Definition 2.4.1)

S[γ r,s ](X, q) = ≡s (mod q) | |≤2|r|X f ( , r)|I(X, , r)| + O r X q X 2/3+
= S[γ r,s ](X, q) + O r X 5/3+ q .

From Proposition 2.4.6, we deduce the equality

S[γ r,s ](X, q) = 6 π 2 2 1 + 1 q 2 (q 2 -2) -1 X 2 q + O ,r q 1+ + X 1/2 q 1/2 (log q) -δ + X 5/3+ q . (2.64)
By the definition (2.14) for C(q), one can easily see that

6 π 2 2 1 + 1 q 2 (q 2 -2) -1 = C(q) 2 + O 1 q 2 .
In conclusion, we proved Proposition 2.5.1. Let C(q) be as in (2.14). There exists δ > 0 such that for every > 0 and every r = 0, one has the asymptotic formula

S[γ r,s ](X, q) = C(q) 2 X 2 q + O ,r q 1+ + X 1/2 q 1/2 (log q) -δ + X 5/3+ q + X 2 q 3 , (2.65)
uniformly for X ≥ 2, for integers s and prime numbers q such that q rs and q ≤ X.

Introduction L Λ (s, χ) = L (s, χ) L(s, χ) and L µ 2 (s, χ) = L(s, χ) L(s, χ 2 ) ,
where L(s, χ) is the classical Dirichlet L-function. In that context, H. L. Montgomery stated two conjectures whose implications are much deeper than the generalized Riemann Hypothesis for Dirichlet L-functions (GRH). The original conjectures can be found, respectively in [31, Formula (15.9), page 136] and [7, top of page 145]. We state them in slightly improved forms, which, in the case of the van Mangoldt function, is due to Friedlander and Granville [START_REF] Friedlander | Limitations to the equi-distribution of primes[END_REF] and in the case of squarefree numbers can be found in a recent preprint by Le Boudec [START_REF] Boudec | On the distribution of squarefree integers in arithmetic progressions[END_REF].

Conjecture 3.1.1. Let , X > 0. Let a and q be integers such that (a, q) = 1 and 1 ≤ q ≤ X, then we have -for primes,

E Λ (X, q, a) = O X X q 1 2 , ( 3.3)
-for squarefree numbers,

E µ 2 (X, q, a) = O X X q 1 4 , (3.4)
where the implied constants depend at most on .

Concerning (3.
3), it is not known, for the moment, if there exists δ > 0 such that

E Λ (X, q, a) = O X q 1-δ
holds in any range whatsoever. For instance, GRH would imply that for every α < 1 2 , there exists δ = δ(α) > 0, such that

E Λ (X, q, a) = O X q 1-δ ,
uniformly for 1 ≤ q ≤ X α and (a, q) = 1. Still under GRH, Turán [START_REF] Turán | Über die Primzahlen der arithmetischen Progression[END_REF] proved that (3.3) holds on average : Uniformly for 1 ≤ q ≤ X, we have

q-1 a=0 (a,q)=1 E Λ (X, q, a) 2 X(log X) 4 .
If one seeks for unconditional results, one needs an extra sum over q ≤ Q, for a certain Q ≤ X.

In this direction, Montgomery proved that for every A > 0, one has

q≤Q q-1 a=0 (a,q)=1 E Λ (X, q, a) 2 = QX log X + O A QX log 2X Q + QX(log X) -A , (3.5)

ON TWO CONJECTURES CONCERNING SQUAREFREE NUMBERS IN ARITHMETIC PROGRESSIONS 75

uniformly for Q ≤ X, where the implied constant depends at most on A. Therefore, for every A > 0, (3.5) implies that (3.3) holds true on average over q ≤ X(log X) -A and a modulo q, with (a, q) = 1.

In the case of squarefree numbers, Prachar [START_REF] Prachar | Über die kleinste quadratfreie Zahl einer arithmetischen Reihe[END_REF] proved that for every > 0, uniformly for (a, q) = 1, X ≥ 2, E µ 2 (X, q, a) X

1 2 + q -1 4 + q 1 2 + ,
where the implied constant depends at most on . This result was later improved by Hooley [START_REF] Hooley | A note on square-free numbers in arithmetic progressions[END_REF] who showed that the above error term can be replaced by

O X 1 2 q -1 2 + q 1 2 + . (3.6)
Both this results show that for every > 0, there exists δ = δ( ) > 0

E µ 2 (X, q, a) X q 1-δ , uniformly for q ≤ X 2 3
-. In a parallel to Túran's result, Corollary 1.1.2 gives the asymptotic formula

q-1 a=0 (a,q)=1 E µ 2 (X, q, a) 2 ∼ C p|q 1 + 2 p 2 -1 X 1 2 q 1 2 ,
as X, q → ∞ and q satisfying X 31 41 + ≤ q ≤ X 1-, where C is an absolute positive constant. This implies (3.4) on average on the above mentioned range. In a subsequent work, le Boudec [START_REF] Boudec | On the distribution of squarefree integers in arithmetic progressions[END_REF] proves that if one seeks for an upper bound rather than an asymptotic formula, one gets a larger range. He proved that

q-1 a=0 (a,q)=1 E µ 2 (X, q, a) 2 X 1 2 + q 1 2 , uniformly for X 1 2 ≤ q ≤ X.
Suppose q ≤ X 1 2 . In this case Hooley's result shows an approximation of (3.4), with 1/2 instead of 1/4. In this paper, we improve the exponent 1/2 for X ≤ q ≤ X 1 2 -, therefore making a further step towards (3.4). For simplicity, we shall henceforth write E(X, q, a) instead of E µ 2 (X, q, a). Theorem 3.1.2. For every > 0, we have E(X, q, a) X q

1 2 + q -1 192 + X q 24 49 + q 3 196 ,
uniformly for every X > 1, every prime number q such that q ≤ X, and every integer a such that (a, q) = 1.

Introduction

Note that for X α < q < 2X α , α ≤ 2 5 , Theorem 3.1.2 shows that E(X, q, a) X q

1 2 - α 192(1-α) + + X q 1 2 -2-5α 196(1-α) +
.

For example, when α = 96 283 < 2 5 , the exponent is 93 187 < 1 2 , which is the best exponent given by Theorem 3.1.2. Therefore, we have the following corollary Corollary 3.1.3. For every 0 < < 1 5 , let δ( ) = min 192(1-) , 25 196(3+5 ) . Then we have

E(X, q, a) X q 1 2 -δ( ) ,
uniformly for every X > 1 and every prime number q such that X ≤ q ≤ X 2 5 -, and every integer a such that (a, q) = 1.

Our next result states that we can still obtain an exponent below 1 2 for q as large as X 1 2 -, but one cannot quantify the exponent in this case Theorem 3.1.4. For every 0 < η < 1 4 , there exists δ = δ(η) > 0 such that we have

E(X, q, a) η X q 1 2 -δ
, uniformly for every X > 1 and every prime number q such that X η ≤ q ≤ X 1 2 -η , and integer a such that (a, q) = 1.

The main new input in the proofs of Theorems 3.1.2 and 3.1.4 are bounds for exponentials sums twisted by the Möbius function given by Fouvry et al. [START_REF] Fouvry | Algebraic trace functions over the primes[END_REF] and Bourgain [3]. The same exponential sums were estimated trivially in [START_REF] Hooley | A note on square-free numbers in arithmetic progressions[END_REF].

Range of uniformity

Concerning the range of uniformity, it is largely believed that if f is sufficiently reasonable, then (3.1) should hold uniformly for q ≤ X 1-. In the case where f = µ 2 , we know, thanks to Prachar [START_REF] Prachar | Über die kleinste quadratfreie Zahl einer arithmetischen Reihe[END_REF] that this is true in the range q ≤ X 2 3 -. Our next result proves that we can overcome the threshold X 2 3 by a small power of log X. More precisely, we prove Theorem 3.1.5. For every 0 < γ < 1 2 , there exists C(γ) such that for every X > 3, for every prime q, for every a coprime with q, one has the inequality

|E(X, q, a)| ≤ C(γ) X 1 3 (log log X) 11 6
(log X)

1 6 -γ 3 + X log log X q(log X) γ 2
.

Perhaps more interesting is the following corollary (obtained by choosing γ = 1-6α 5 in Theorem 3.1.5 above) : 3.2.1 -Approximation to the ψ function Corollary 3.1.6. For every 0 < α < 1 6 , there exists C(α) such that for every X > 3, for every prime q such that q ≤ X 2 3 (log X) α , for every a coprime with q, one has the inequality

|E(X, q, a)| ≤ C(α) X(log log X) 11 6
q(log X)

1-6α 10

.

In particular, for every constant > 0, the asymptotic formula

n≤X n≡a (mod q) µ 2 (n) ∼ 1 ϕ(q) n≤X (n,q)=1 µ 2 (n) ∼ 6 π 2 1 - 1 p 2 -1 X q holds as X → ∞, uniformly for q ≤ X 2 3 (log X) 1 6
and integers (a, q) = 1. This is the first time ocurrence of such an asymptotic formule for prime values of q tending to infinity faster than X 2 3 .

Finally, We remark that Corollary 3.1.6 implies that if n(a, q) denotes the least positive squarefree number that is congruent to a modulo q, then for every > 0, we have the inequality

n(a, q) q 3 2 (log q) -1 6 + ,
where the implied constant depends only on . The best result in this direction is due to Heath-Brown [START_REF] Heath-Brown | The least square-free number in an arithmetic progression[END_REF], who proved that n(a, q) (d(q) log q) 6 q 13 9 .

Preliminary results

The next lemma is a simple consequence of Weil's bound for exponential sums that come from algebraic curves over finite fields and classical estimates for Gauss sums. Lemma 3.2.1. Let A < B be real number, let a and q be integers satisfying (a, q) = 1, q ≥ 1. Then for every > 0, we have A<r≤B e ar 2 q q B -A q

1 2 + q 1 2
where r denotes the multiplicative inverse of r (mod q), e(z) = e 2πiz and the implicit constant depends at most on .

Approximation to the ψ function

The next lemma is an useful analytic tool to avoid the problems arising from the lack of continuity of the sawtooth function. The version we use here can be found in [START_REF] Fouvry | On a theorem of Bombieri-Vinogradov type[END_REF], and is inspired by an idea of Vinogradov.

Preliminary results

Lemma 3.2.2. (see [START_REF] Fouvry | On a theorem of Bombieri-Vinogradov type[END_REF]Lemma 4]) Let ψ(x) = x-x -1 2 and Y > 1. There are two functions A and B with period 1 such that, for every real x, one has

|ψ(x) -A(x)| ≤ B(x), where A(x) = h =0
A h e(hx),

B(x) = Y -1 + h =0 B h e(hx), with A h , B h C h := min 1 |h| , Y 3 |h| 4 , (h = 0). (3.7)

Exponential sums twisted by the Möbius function

We now state the estimates for exponential sums twisted by the Möbius functions that were mentioned in the introduction. The first one is a very particular case of [START_REF] Fouvry | Algebraic trace functions over the primes[END_REF]Theorem 1.7] by Fouvry, Kowalski and Michel, which was based on a previous work by Fouvry and Michel [START_REF] Fouvry | Sur certaines sommes d'exponentielles sur les nombres premiers[END_REF]. It gives non-trivial bounds for R ≥ q 3 4 + and will be used in several places of the proof of Theorem 3.1.2. Theorem 3.A. For every > 0, there exists C( ) such that, for every R ≥ 1, for every prime q, and every a coprime with q, one has the inequality

n≤R µ(n)e an 2 q ≤ C( )R 1 + q R 1 12 q -1 48 + . (3.8)
To prove Theorem 3.1.4, we need to replace Theorem 3.A by an estimate that gives something non-trivial in the larger range R ≥ q 1 2 + . For this we have the next result which is a combination of a remarkable result by Bourgain [3], which is non-trivial in the range q 1 2 + ≤ R ≤ q, and Theorem 3.A itself. Theorem 3.B. For every η > 0, there exists δ(η) > 0 and C(η) such that, for every R ≥ 1, for every prime q satisfying q 1 2 +η ≤ R ≤ q 1 η , and every a coprime with q, one has n≤R µ(n)e an 2 q ≤ C(η)R 1-δ(η) .

(3.9)

Démonstration. For q 1 2 +η ≤ R ≤ q, this is exactly [3, Theorem A.9]. For q < R ≤ q 1 2η , it follows from Theorem 3.A.

-Selberg's Sieve

Short exponential sums

In the course of the proof of Theorem 3.1.5, we are led to deal with very short exponential sums, for which we use the following result by Bourgain-Garaev. It gives non-trivial results for short Kloosterman sums (mod q) where the length is as small as q for any > 0.

Theorem 3.C. (see [5,Theorem 16]) There exists an absolute constant C such that for every M ≥ 2, every prime q, and every a coprime with q, one has the inequality

m≤M e am q ≤ C M log q(log log q) 3 (log M ) 3 2
,

The following lemma is obtained by combining Lemma 3.2.2 and Theorem 3.C above.

Lemma 3.2.3. Let ψ(x) be as in Lemma 3.2.2. Then we have the inequality

m≤M ψ N + am q M log q + M log q(log log q) 4 (log M ) 3 2
.

uniformly for every pair of real numbers M, N such that M > 1 and prime q > 2, where the implied constant is absolute.

Démonstration. Let 1 ≤ Y ≤ q to be chosen later. By Lemma 3.2.2, we deduce that

m≤M ψ N + a m q M Y + M ∞ h=-∞ h =0 C hq + q h C h m≤M e ah m q ,
where C h is as in (3.7). By Theorem 3.C and the bounds (3.7), we have that

m≤M ψ N + a m q M Y + M log q(log log q) 3 (log M ) 3 2 log Y, since Y ≤ q.
We conclude by choosing Y = log q.

Selberg's Sieve

Another important input to the proof of Theorem 3.1.5 is the Selberg sieve for detecting squares (cf. [START_REF] Friedlander | Opera de cribro[END_REF]Chapter 8]). We shall need the following result. Theorem 3.D. Let A = (a n ) be a finite sequence of non-negative numbers. Let P be a squarefree number. For each p | P , let Ω p be a set of congruence classes (mod p). For every d | P we write

|A d | = n (mod d)∈Ωp
for every p|d

a n = g(d)Y + r d (A), (3.10)
where Y > 0 and g(d) is a multiplicative function with 0 < g(p) < 1 for p | P . Let h(d) be the multiplicative function given by h(p) = g(p)(1 -g(p)) -1 and for any D > 1 define

J = J(D) := d|P d< √ D h(d).
Then, for any D > 1 we have the inequality

n (mod d) ∈Ωp for every p|P a n ≤ Y J -1 + d|P d≤ √ D τ 3 (d) |r d (A)| ,
where τ 3 is the generalized divisor function. The optimal choice of these ρ d is the heart of the Selberg's sieve.

Proofs of the results

Proof of Theorem 3.1.2

Let X > 1 and > 0 be real numbers and let q be a prime number. Since the upper bound given by Theorem 3.1.2 is worse than (3.6) for q ≤ X 2/5 , we may suppose q ≤ X 2/5 . Let

S := (r,q)=1 µ(r) m≤X/r 2 m≡ar 2 (mod q) 1, (3.11)
and

S 0 := 1 ϕ(q) n≤X (n,q)=1 µ 2 (n), (3.12)
We have E(X, q, a) = S -S 0 .

(3.13)

3.3.1 -Proof of Theorem 3.1.2 81
It is rather elementary to see that S 0 satisfies (recall that q is prime)

S 0 = 6 π 2 1 - 1 q 2 -1 X q + O (X 1 2 q -1+ ). (3.14) Let 1 < R ≤ X 1 2
be a parameter to be chosen later depending on X and q. We split S as

S = S I + S II , (3.15)
where

               S I = r≤R (r,q)=1 µ(r) m≤X/r 2 m≡ar 2 (mod q)
1,

S II = R<r≤X 1 2 (r,q)=1 µ(r) m≤X/r 2 m≡ar 2 (mod q) 1. (3.16)
For the first sum in (3.16), we have that

S I = r≤R (r,q)=1 µ(r) X qr 2 -ψ X qr 2 - ar 2 q + ψ - ar 2 q =: T -U + V, (3.17)
say. The first term satisfies

T = 6 π 2 1 - 1 q 2 -1 X q + O R -1 Xq -1 . (3.18)

Study of V

Let 1 < Y ≤ X be a parameter to be chosen optimally later depending on X and q, then Lemma 3.2.2 gives us two functions A and B whose Fourier coefficients satisfy (3.7), and such that

|V| ≤ V 1 + V 2 , (3.19)
where

V 1 = r≤R (r,q)=1 µ(r)A - ar 2 q , V 2 = r≤R (r,q)=1 B - ar 2 q .
Writing down the Fourier development for A(x), and using (3.7), we see that Proofs of the results

V 1 ≤ h =0 C h r≤R (r,q)=1 µ(r)e - ahr 2 q . (3.20)
The contribution of the terms where q | h is trivially seen to be

R h =0 C hq X Rq -1 , (3.21)
by (3.7). For the remaining terms, we use Theorem 3.A and see that their contribution is

X Rq -1 48 + R 11 12 q 1 16 , (3.22)
again by (3.7). Hence, by (3.20), (3.21) and (3.22), we have

V 1 X Rq -1 48 + R 11 12 q 1 16 . (3.23)
The analysis of V 2 is completely analogous. The only difference is that we shall need Lemma 3.2.1 instead of Theorem 3.A. We obtain

V 2 X Rq -1 2 + q 1 2 + RY -1 . (3.24)
Gathering (3.23) and (3.24) in (3.19), we have that

V X Rq -1 48 + R 11 12 q 1 16 + q 1 2 + RY -1 . (3.25)

Study of U

This part is very similar to the study of V but with the difference that we need an Abel summation to take care of the oscillation of the term X/qr 2 . Let 1 < R 0 ≤ (X/q) 1 2 to be chosen optimally later. We write

U = W + O(R 0 ), (3.26)
where

W := R 0 <r≤R (r,q)=1 µ(r)ψ X qr 2 - ar 2 q .
Again by Lemma 3.2.2, we obtain two functions A and B satisfying (3.7) and such that

|W| ≤ W 1 + W 2 , (3.27)
where

3.3.1 -Proof of Theorem 3.1.2 83 W 1 = R 0 <r≤R (r,q)=1 µ(r)A X qr 2 - ar 2 q , W 2 = R 0 <r≤R (r,q)=1 B X qr 2 - ar 2 q .
We write down the Fourier development of A(x) and again, we separate the contribution from the terms where q | h as we did for V 1 . We deduce

W 1 ≤ (h,q)=1 C h R 0 <r≤R (r,q)=1 µ(r)e hX qr 2 - ahr 2 q + O X Rq -1 . (3.28)
Summing by parts, we see that

R 0 <r≤R (r,q)=1 µ(r)e hX qr 2 - ahr 2 q |h|X q R 0 <t≤R (t,q)=1 1 t 3 S(t, q) + S(R, q) + S(R 0 , q),
where

S(t, q) := max (a,q)=1 r≤t (r,q)=1 µ(r)e ar 2 q (3.29)
The terms on the right-hand side of the above inequality can be estimated by means of Theorem 3.A giving

R 0 <r≤R (r,q)=1 µ(r)e hX qr 2 - ahr 2 q X |h|R -1 0 Xq -49 48 + |h|R -13 12 0 Xq -15 16 +Rq -1 48 + R 11 12 q 1 16 + R 0 .
Injecting it in (3.28) and using (3.7), we deduce

W 1 X R -1 0 XY q -49 48 + R -13 12 0 XY q -15 16 + Rq -1 48 + R 11 12 q 1 16 + R 0 . (3.30)
The treatment of W 2 goes in a similar fashion, replacing Theorem 3.A by Lemma 3.2.1 in the appropriate places. We end up with

W 2 X R -1 0 XY q -3 2 + R -2 0 XY q -1 2 + Rq -1 2 + q 1 2 + RY -1 + R 0 . (3.31)
Gathering (3.30) and (3.31) in (3.27), we have that 84 Proofs of the results

W X R -1 0 XY q -49 48 + R -13 12 0 XY q -15 16 + Rq -1 48 + R 11 12 q 1 16 + R -2 0 XY q -1 2 +q 1 2 + RY -1 + R 0 . (3.32)
Putting together (3.17), (3.18), (3.25), (3.26) and (3.32), we see that

S I = 6 π 2 1 - 1 q 2 X q + O X R -1 Xq -1 + Rq -1 48 + R 11 12 q 1 16 + q 1 2 + RY -1 +R -1 0 XY q -49 48 + R -13 12 0 XY q -15 16 + R -2 0 XY q -1 2 + R 0 . (3.33)

Study of S II

We procceed now to estimate S II . We follow the lines of [23, Lemma 2]. Ignoring the oscillation of µ, we see that

S II R<r≤ √ X m≤ X r 2 r 2 m≡a (mod q) log 3 X r 2 m = R<r≤ √ X m≤ X r 2 r 2 m≡a (mod q) 3 X m r dt t 3 √ X R m≤ 9X t 2 r≤t r 2 m≡a (mod q) 1 dt t . (3.34)
We put

Z = max(R, q) (3.35)
and break up the integral on the right-hand-side from (3.34) as

Z R m≤ 9X t 2 r≤t r 2 m≡a (mod q) 1 dt t + 3 √ X Z m≤ 9X t 2 r≤t r 2 m≡a (mod q) 1 dt t
For the first integral, we use additive characters to detect the congruence condition. We have

m≤ 9X t 2 r≤t r 2 m≡a (mod q) 1 = 1 q 2 q-1 α=0 q-1 β=0 S(q; α, aβ)Θ(t, α)Θ 9X t 2 , β , (3.36)
where

S(q; α, β) := q-1 h=1 e αh + β h2 q , and
Θ(t, α) := n≤t e - αn q min t, α q -1 . (3.37)
In order to estimate the sum on the right-hand side of (3.36), we need bounds for S(q; α, β). In the cases where αβ ≡ 0 (mod q), the sum is either trivial, a Ramanujan sum or a Gauss sum. And the classical upper-bound for these sums are used. If both α and β are ≡ 0 (mod q), then we shall use the following upper-bound that follows from the work of Weil S(q; α, β) q 1 2 , (α, β ≡ 0 (mod q)).

Combining these bounds with (3.37), we see that

m≤ 9X t 2 r≤t r 2 m≡a (mod q) 1   X qt + t q 3 2 q-1 β=1 β q -1 + X q 3 t q-1 α=1 α q -1 + 1 q 3 2 q-1 α=1 q-1 β=1 α q -1 β q -1   X X qt + t q 1/2 + q 1/2 (3.38)
Notice that if Z = R, this first integral vanishes. So we can suppose Z = q. With that in mind, if we integrate both sides of inequality (3.38) against dt t , we obtain

Z R m≤ 9X t 2 r≤t r 2 m≡a (mod q) 1 dt t X X Rq + q 1/2 . (3.39)
For the remaining integral, we notice that for fixed m, the equation r 2 m ≡ a (mod q) has at most two solutions for r (mod q) (recall that q is prime). Thus, we have (since

Z ≥ q) 3 √ X Z m≤ 9X t 2 r≤t r 2 m≡a (mod q) 1 dt t X Zq ≤ X Rq . (3.40)
Adding up (3.39) and (3.40), we have, in view of (3.34), that

S II X R -1 Xq -1 + q 1/2 . ( 3
S - 6 π 2 1 - 1 q 2 -1 X q X R -1 Xq -1 + Rq -1 48 + R 11 12 q 1 16 + q 1 2 + RY -1 +R -1 0 XY q -49 48 + R -13 12 0 XY q -15 16 + R -2 0 XY q -1 2 + R 0 . (3.42)
Forcing the first, the fifth and the last terms to be equal, we are faced with the choices

R = X q 1 2 Y 1 2 , R 0 = X q 1 2 Y -1 2 .
Injecting these values in (3.42), we see that

S - 6 π 2 1 - 1 q 2 -1 X q X X q 1 2 Y -1 2 + X q 1 2 Y 3 2 q -1 48 + X q 11 24 Y 37 24 q 1 16 + Y 2 q 1 2 . (3.43) Take Y = min q 1 96 , X 1 49 q -5 98 , X 1 5 q -2 5
to optimize the right-hand side of (3.43). Then (3.13), (3.14) and (3.43) imply that we have E(X, q, a) X X q

1 2 q -1 192 + X q 24 49 q 3 196 + X q 2 5 q 1 10
, and the last term is dominated by the first in the range q ≤ X 2 5 . Hence we conclude the proof of Theorem 3.1.2.

Proof of Theorem 3.1.4

Let X > 1 and η > 0 be real numbers and let q be a prime number such that X η ≤ q ≤ X 1 2 -η . We let again S and S 0 be as in the previous section (see (3.11) and (3.12)). Notice that we have X q

1 2 ≥ q 1 2 +η .
Let δ 1 > 0 to be chosen later depending on η. Also let

R = X q 1 2 +δ 1 , R 0 = X q 1 2 -δ 1 , Y = X q 2δ 1 . (3.44)
Notice that we can choose δ 1 sufficiently small so that

R 0 ≥ q 1 2 + η 2 .
Theorem 3.B now gives us a certain δ 2 > 0 depending on η such that

S(t, q) ≤ t 1-δ 2 , (t ≥ R 0 ), (3.45)
where S(t, q) is as in (3.29). We start as in the last section, writing

S = S I + S II , (3.46)
where S I and S II are as in (3.16). We deal S I in the exact same way as before, only replacing each use of Theorem 3.A by the upper bound (3.45). Thus we obtain

S I = 6 π 2 1 - 1 q 2 X q + O ,η X R -1 Xq -1 + R 1-δ 2 + RY -1 + R -1-δ 2 0 XY q -1 +R -1 0 XY q -3 2 + R -2 0 XY q -1 2 + R 0 , (3.47)
for any > 0 (compare with (3.33)).

As for S II , we have the exactly same bound as in the previous case (see (3.41)). Gathering (3.41), (3.47) and (3.46), we see that

S - 6 π 2 1 - 1 q 2 X q ,η X R -1 Xq -1 + R 1-δ 2 + RY -1 + R -1-δ 2 0 XY q -1 +R -1 0 XY q -3 2 + R -2 0 XY q -1 2 + R 0 .
Now, we deduce from (3.13), (3.14) and (3.44) the inequality (recall that X

η ≤ q ≤ X 1 2 -η ) E(X, q, a) ,η X   X q 1 2 -δ 1 + X q 1 2 +3δ 1 - δ 2 2 +δ 1 δ 2 + X q 1 2 +3δ 1 -η 2 + X q 1 2 +4δ 1 -2η   .
Notice that since q ≤ X 1/2 , one has X ≤ (X/q) 2 . Now, taking < δ 1 /4 and δ sufficiently small, we deduce

E(X, q, a) η X q 1 2 - δ 1 2
.

Taking δ := δ 1 /2 concludes the proof of Theorem 3.1.4.

Proof of Theorem 3.1.5

Let X > 1 and let q be a prime number such that q ≤ X. Since the upper bound from Theorem 3.1.5 is worse than (3.6) for q ≤ X 1 2 , we can suppose that q ≥ X 1 2 . Let S and S 0 be as in (3.11) and (3.12), respectively, and 1 < R ≤ X 1 3 be a parameter to be chosen optimally later. We split S as before, writing

S = S I + S II , (3.48)
where S I and S II are as in (3.16). For S I , it suffices to detect the congruence trivially. We have

S I = r≤R (r,q)=1 µ(r) X qr 2 + O(1) = 6 π 2 p|q 1 - 1 p 2 -1 X q + O(R + R -1 Xq -1 ). (3.49)
For S II , by estimating the µ function trivially and changing the order of summation, we have that

|S II | ≤ m≤X/R 2 r≤(X/m) 1/2 r 2 m≡a (mod q) 1 = S III + S IV , (3.50)
where

               S III = m≤ X R 2 (log X) r≤(X/m) 1/2 r 2 m≡a (mod q) 1, S IV = X R 2 (log X) <m≤ X R 2 r≤(X/m) 1/2 r 2 m≡a (mod q) 1. (3.51)
For S III , we detect the congruence trivially, obtaining

S III = m≤ X R 2 (log X) X 1 2 m 1 2 q + O(1) X Rq(log X) 1 2 + X R 2 (log X) . (3.52)
As for S IV , we proceed by dyadic decomposition of the values of m. Doing so, we find that there exists M ≥ 1, a power of two, such that

X 2R 2 (log X) < M ≤ 2X R 2 , (3.53)
and

S IV S log log X, (3.54)
where S = S(X, M ; q, a) is given by

S := M <m≤2M r≤(X/M ) 1/2 r 2 m≡a (mod q) 1. (3.55)
In what follows, we show how to use the Selberg's sieve (see Theorem 3.D) to estimate S.

Implementing the Selberg's sieve

Let

a n = M <m≤2M mn≡a (mod q) 1, if n ≤ X M , (3.56)
and a n = 0, otherwise. Then

S = n= a n ,
where the condition n = means that we only sum over the n that are perfect squares.

Let P be a product of distinct odd primes such that p q. For each p | P , let Ω p denote the set of non-square residue classes (mod p). Note that we can soften the condition n = to n ∈ Ω p for every p | P . In other words, the following inequality holds :

S ≤ n ∈Ωp for every p|P a n .
We want to use Theorem 3.D. Thus, we need to give asymptotic formulas for |A d | (see (3.10)). We notice that with a n as in (3.56), we have

G (d, α) = M <m≤2M X M dq -ψ X M dq - adm q - αq d + ψ - adm q - αq d ,
for every d coprime with q and α (mod d). By Lemma 3.2.3, we obtain that

G (d, α) = X dq + O M log q(log log q) 4 (log M ) 1 2
.

(3.59)

Note that the inequality (3.53) implies

M ≥ X 1 3 (log X) -1 .
Hence, (3.57) and (3.59) imply

|A d | = g(d) X q + O g(d)d M (log log X) 4 (log X) 1 2 , (3.60)
where g(d) is the multiplicative function supported on squarefree numbers and such that g(p) = p -1 2p .

Let D > 1 and

P := 2<p≤ √ D p q p.
We use Theorem 3.D for Y = X q and D and P as above. We obtain, in view of (3.60), the inequality

S ≤ n ∈Ωp for every p|P a n X q J -1 + M (log log X) 4 (log X) 1 2 d≤ √ D τ 3 (d)g(d)d,
where (recall that q is prime)

J = d< √ D d|P
+ DX(log D) 1 2 (log log X) 5 R 2 (log X) 1 2
.

For each 0 < γ < 1 2 , the choice D = (log X) γ gives the inequality S IV X log log X q(log X)

γ 2 + X(log log X) 11 2 R 2 (log X) 1 2 -γ . ( 3
S - 6 π 2 1 - 1 q 2 -1 X q R + R -1 Xq -1 + X log log X q(log X) γ 2 + X(log log X) 11 2 R 2 (log X) 1 2 -γ . (3.63)
Forcing the first and last terms to be equal, we are faced with the choice

R = X 1 3 (log X) -1 6 + γ 3 (log log X) 11 6
.

Replacing it in (3.63) gives the inequality

S - 6 π 2 1 - 1 q 2 -1 X q X 1 3 (log log X) 11 6
(log X)

Chapitre 4

On a result of R. R. Hall on squarefree numbers in short intervals

Introduction

Let µ denote, as usual, the Möbius function, so that µ 2 is the arithmetic function which detects the squarefree numbers. The sequence of squarefree numbers has a natural density of 6 π 2 . Indeed, one has the classical asymptotic formula with a power saving error term

n≤X µ 2 (n) = 6 π 2 X + O( √ X), uniformly for X > 1.
Under the Riemann hypothesis, the formula would hold with an even smaller error term. The best conditional estimate for the error term was obtained by Jia [START_REF] Jia | The distribution of squarefree numbers[END_REF], who proves that we can replace O( √ X) by O (X 17/54+ ), for every > 0.

Short intervals

If we switch to the question of squarefree numbers lying in short intervals, much less is known. The question was addressed by Hall in [START_REF] Hall | The distribution of squarefree numbers[END_REF], where he studied the following counting function

N (n, k) := 0≤h<k µ 2 (n + h),
for n and k positive integers. In view of the density of squarefree numbers, we may hope that we have N (n, k) ∼ 6 π 2 k as n and k tend to infinity, with a large uniformity on k and n. Thus it is natural to define the discrepancy D(n, k) := N (n, k) -6 π 2 k. Hall's main goal was to give an estimate, for X > 0, of the following (upper) density

d k (X) = lim sup x→∞ 1 x # n ≤ x; |D(n, k)| > X ( ≥ 1).
In order to establish his result, he considered the moments

M(x, k; ) := 0<n≤x D(n, k) . (4.1)
Expanding D(n, k) , he was immediately led to consider the sum

S(x; h) := n≤x µ 2 (n + h 1 ) . . . µ 2 (n + h ), (4.2)
where h = (h 1 , . . . , h ) is a fixed -tuple of nonnegative integers. He recovers the asymptotics (see also [START_REF] Mirsky | Arithmetical pattern problems relating to divisibility by rth powers[END_REF])

S(x; h) ∼ A (h) x, as x → ∞, (4.3)
where

A (h) = p 1 - u p (h) p 2 , (4.4)
and u p (h) = # h 1 , . . . , h (mod p 2 ) ,

i.e. u p (h) counts the number of distinct residue classes (mod q) covered by h 1 , . . . , h . Each factor on the right side of (4.4) is the probability of the event p 2 n + h 1 , . . . , p 2 n + h , and (4.3) states that these events are (asymptotically) independent. From the definition (4.1) and the asymptotic formula (4.3), Hall deduces the asymptotic formula (for fixed integers k and ≥ 1) :

M(x, k; ) ∼ C (k)x, as x → ∞, (4.6)
where

C (k) := j=0 j - 6 π 2 k -j B j (k), (4.7)
with B 0 (k) := 1, and, for j ≥ 1,

B j (k) := k-1 h 1 =0 . . . k-1 h j =0 A(h). (4.8)
Since |A(h)| ≤ 1, the trivial bound is

C (k) k , uniformly for k ≥ 1, (4.9)
which is certainly far from the true order of magnitude since one expects cancellations due to the sign changes in formula (4.7). Among other results, Hall proved (see [START_REF] Hall | Squarefree numbers on short intervals[END_REF] and [START_REF] Hall | The distribution of squarefree numbers[END_REF]) the following improvements on the trivial bound (4.9)

C 2 (k) = Ck 1/2 + O (k 1/3+ ), (4.10)
for every > 0, as k → ∞, where

C = ζ 3 2 πζ(2) p p 3 -3p + 2 p 3 = 0, 167 . . . ; (4.11)
and, for each ≥ 3,

C (k) k ( -1)/2 , (4.12)
uniformly for k ≥ 1. This has obvious consequences to the moments M (x, k), or even to the general absolute moments

M + (x, k; λ) := 0<n≤x |D(n, k)| λ , (4.13)
where λ is a positive real number. Indeed, the upper bound (4.12), the asymptotic formula (4.6) and Hölder's inequality imply

lim inf x→∞ 1 x M + (x, k; λ) λ k λ/4 , λ ≥ 2, k (λ-1)/2 , 1 ≤ λ < 2.
and

lim sup x→∞ 1 x M + (x, k; λ) λ k (λ-1)/2 , λ ≥ 2, k λ/4 , 1 ≤ λ < 2.
Inspired by the results from Chapter 1, we are able to improve (4.12) following a somewhat involved but rather elementary method. We now state our results

Theorem 4.1.1. For each integer ≥ 3, we let

θ :=          2 3 + 2 , for = 3, 4 -2 2 , for ≥ 5, (4.14)
and let C (k) be defined by (4.7). Then for every > 0 and every ≥ 3 there exists C , > 0 such that for every k ≥ 1, we have

|C (k)| ≤ C , k θ + . (4.15)
As a direct consequence of Theorem 4.1.1 and Theorem 4.6.1 (cf. Section 4.6 below), which is an effective version of (4.3), we have the following. Corollary 4.1.2. Let M(x, k; ) be defined as in (4.1). Let θ be as in (4.14). For every > 0 and every ≥ 3 there exists D , > 0 such that for every 1 ≤ k ≤ x, we have

|M(x, k; )| ≤ D , k θ + x + k x +1 + , (4.16)
In particular, for every > 0, for every ≥ 3, for every 1 ≤ k ≤ x δ -, we have the inequality

|M(x, k; )| ≤ 2D , k θ + x,
where δ := 1 ( + 1)( -θ ) . (4.17)

As in [START_REF] Hall | The distribution of squarefree numbers[END_REF], we can also give estimates for the absolute moments M + (see (4.13)). Indeed we have Corollary 4.1.3. For every , λ > 0, there exists c + ,λ > 0 such that for every positive integer k, we have the inequality

lim inf x→∞ 1 x M + (x, k; λ) ≥ c + ,λ k α λ -,
where

α λ = λ 4 , λ ≥ 2, 9λ-4 28 , 0 < λ < 2.
Similarly, for every , λ > 0, there exists C + ,λ > 0 such that for every positive integer k, we have the upper bounds lim sup

x→∞ 1 x M + (x, k; λ) ≤ C + ,λ k β λ + ,
where

β λ =            λ-2 2 , λ ≥ 6, 3λ-4 7 , 4 ≤ λ < 6. 9λ-4 28 , 2 ≤ λ < 4. λ 4 , 0 < λ < 2.
Démonstration. Suppose 0 < λ ≤ 2. By Hölder's inequality, we have both This improves on the similar bound from [START_REF] Hall | The distribution of squarefree numbers[END_REF], where 5 28 is replaced by 0.

M(x, k; 2) ≤ M + (x, k; λ)

Arithmetic progressions

The distribution of arithmetic sequences in short intervals shares many similarities with the distribution of these sequences in arithmetic progressions. The deepest example concerns the sequence of primes. In the following we introduce the basic elements to the study of squarefree numbers in arithmetic progressions, trying to emphasize as much as possible the origin of this duality, at least in the current case.

Let X > 1 and let q be a positive integer. It is easy to prove that the number of squarefree numbers ≤ X which are coprime with q is asymptotically equivalent to

6 π 2 p|q 1 - 1 p 2 -1 ϕ(q) q X.
The question of whether these numbers are well distributed among the arithmetic progressions (mod q) amounts to study the following error terms. Let a be an integer such that (a, q) = 1.

Then we define the error term E(X, q, a) by the following equation

n≤X n≡a (mod q) µ 2 (n) = 6 π 2 p|q 1 - 1 p 2 -1 X q + E(X, q, a).
It is clear that E(X, q, a) satisfies E(X, q, a) ≤ X q + 1.

We are interested in finding regions for X and q in which E(X, q, a) = o (X/q) . (4. [START_REF] Hall | The distribution of squarefree numbers[END_REF] In Chapter 4, we proved that for every > 0, (4.19) holds true as X → ∞, for arbitrary q ≤ X 2 3 (log X) 6and (a, q) = 1, thus breaking the barrier of X 2 3 from the results of Prachar [START_REF] Prachar | Über die kleinste quadratfreie Zahl einer arithmetischen Reihe[END_REF] and Hooley [START_REF] Hooley | A note on square-free numbers in arithmetic progressions[END_REF]. We only recall the result of Hooley [START_REF] Hooley | A note on square-free numbers in arithmetic progressions[END_REF], who showed E(X, q, a) = O X q

1/2 + q 1/2+ , (4.20)
for > 0 arbitrary, where the O -constant depends on alone. We believe that (4.19) should hold as X → ∞ for arbitrary q ≤ X 1-and (a, q) = 1. In the region q ≤ X 1/2-, formula uniformly for (a, q) = 1, X θ 1 < q < X θ 2 where the values of the constants θ 1 and θ 2 were not made precise. In Chapter 4, we make progress towards (4.21) by proving that for every > 0 there exists δ = δ( ) > 0 such that for every X ≤ q ≤ X 1 2 -, we have

E(X, q, a) = O X q 1 2 -δ
.

We define the -th moment for this distribution as M(X, q; ) = * a (mod q) E(X, q, a) , (4.22) where the * symbol means that we only sum over the classes that are relatively prime to q.

Inspired by Montgomery's conjecture (cf. (4.21)), we make the following related conjecture for the moments :

Conjecture 4.1.5. For every integer ≥ 1 and every > 0, there exists δ > 0, η > 0 and a positive multiplicative function c such that M(X, q; ) -c (q)ϕ(q) X q

4 ≤ ϕ(q) X q 4 -η , (4.23)
for every q and X satisfying X 1-δ ≤ q ≤ X 1-, X ≥ 2.

Furthermore

-if is even, c is a bounded function, -if is odd, c (q) = 0 for every integer q.

For simplicity of notation in subsequent formulas, we define

A q = 6 π 2 p|q 1 - 1 p 2 -1 . (4.24)
In an analogous way to the case of short intervals, the study of M(X, q; ) depends highly on the behavior of C X q ; q , where C (., .) is the following analogue of C (k). For positive integers and q, and Y ≥ 1, we write

C (Y ; q) := j=0 j (-A q Y ) -j B j (Y ; q), (4.25)
with B 0 (Y ; q) := 1, Introduction and

B j (Y ; q) := 1 0 -u<h 1 ≤Y -u . . . -u<h j ≤Y -u A q (h)du, for j ≥ 1, (4.26)
where

A q (h) := p 1 - u p (h) p 2 p|q 1 - u p (h) p 2 -1 , (4.27)
and the u p (h) are as in (4.5). To illustrate how definition (4.26) generalizes (4.8), we note that if Y = k is an integer and q = 1, then for every u ∈ (0, 1), the condition

-u < h i ≤ k -u is equivalent to 0 ≤ h i ≤ k -1.
Thus we see that

B j (k; 1) = 1 0 0≤h 1 ≤k-1
. . .

0≤h j ≤k-1 A(h)du = B j (k).
Thus we have (note that A 1 = 6 π 2 ), C (k; 1) = C (k), for every positive integer k.

(4.28)

The precise relationship between M(X, q; ) and C X q ; q is given by formula (4.95). This formula is an effective analogue of (4.6) in the case of arithmetic progressions. Identity (4.28) and the similarity between (4.6) and (4.95) are the cornerstone of the analogy between the distributions of squarefree integers in short intervals and in arithmetic progressions. This resemblance had already appeared in [START_REF] Hall | The distribution of squarefree numbers[END_REF]Theorem 1.1], where we proved the formula

M(X, q; 2) = C p|q p 2 + p -2 p 2 -1 X 1 2 q 1 2 + O d(q)X 1 3 q 2 3 + X 23 15 + q -13 15 , (4.29)
where C is as in (4.11).

In view of identity (4.28), Theorem 4.1.1 is a particular case of Theorem 4.1.6. For every > 0 and for every ≥ 3 there exists C , > 0 such that for every Y ≥ 1 and every q ≥ 1, one has the inequality

|C (Y ; q)| ≤ C , Y θ + , (4.30)
where θ is as in (4.14).

Theorem 4.1.6 is our main result and we dedicate a large part of the paper to its proof. In the same spirit of Corollary 4.1.2 (i.e. as a consequence of Theorem 4.6.1), we have the following corollary.

Corollary 4.1.7. Let M(X, q; ) be defined by (4.22). Let θ be as defined in (4.14) and let δ be as in (4.17).Then for every > 0 and every ≥ 3 there exists D , > 0 such that for every X ≥ 1 and every positive integer q ≤ X, we have

|M(X, q; )| ≤ D , ϕ(q) X q θ + + X +1 X q -1 , (4.31)
4.1.3 -Overview of the methods

99

In particular, for every > 0, for every ≥ 3, for every X ≥ 1 and every positive integer q such that X 1-δ + ≤ q ≤ X, we have |M(X, q; )| ≤ 2D , ϕ(q) X q θ + .

Notice that this result makes a step towards Conjecture 4.1.5 for ≥ 3. For example, in the particular case where = 3, we notice that Corollary 4.1.7 gives that for every > 0, we have |M(X, q; 3)| ϕ(q) X q 9 11 + , uniformly for X ≥ 1 and integers q such that X 85 96 + ≤ q ≤ X. This upper bound is quite close to what is predicted by Conjecture 4.1.5, where 9 11 + is replaced by 3 4 -η for some η > 0. We also remark that Hölder's inequality can give similar results to Corollaries 4.1.3 and 4.1.4 in the context of arithmetic progressions. That means that we can get lower and upper bound for the absolute moments defined by M + (X, q; λ) := * a (mod q) |E(X, q, a)| λ .

In the particular case where λ = 3, we have that for every > 0, the inequalities ϕ(q) X q 3 4 M + (X, q; 3) ϕ(q) X q 23 28 +

, hold uniformly for X ≥ 1 and integers q such that X 19 20 + ≤ q ≤ X.

Overview of the methods

As remarked by Hall (see the remark at the bottom of [18, p. 12]), for ≥ 3, obtaining asymptotic formulas for B j (Y ; q) (resp. B j (k)), j = 1, . . . , is not enough to obtain good results for C (Y ; q) (resp. C j (k)). The reason is that many secondary main terms are hidden in B j (Y ; q). Lemma 4.3.5 plays a major role in getting rid of these secondary terms. At this point, we need good estimates for E (Y ; r) (see (4.40)). This comes in the form of Proposition 4.3.8, Lemma 4.4.1 and Lemma 4.4.2.

These results are linked to counting points in boxes of Z with congruence conditions. For example, let r = (r 1 , . . . , r ) be an -tuple of positive integers. We consider

N (Y ; r) := # h ∈ Z ; 0 < h 1 , . . . , h ≤ Y and (r i , r j ) 2 | h i -h j for all 1 ≤ i, j ≤ .
An asymptotic formula for N (Y ; r) is available for small values of the r j . This is, in some sense, captured by Lemma 4.4.1. As for large values of the r j , we do not obtain such an asymptotic formula. We are content with upper bounds. In this context, Proposition 4.3.8 deals with the diagonal terms (when some of the h j are equal) and Lemma 4.4.

E 2 (Y ; r 1 , r 2 ) = 2d 2 Y d 2 0 ψ(u)du,
where d = (r 1 , r 2 ). This identity is essential in obtaining the asymptotic formula (4.29). It would be very interesting to obtain a generalization of such an identity for ≥ 3.

Notation

We

Preparatory results

We start with the following simple lemma Démonstration. Since both sides of (4.33) are invariant by x → x + 1 n , we can restrict ourselves to the case where 0 ≤ x < 1 n . In which case, the left-hand side of (4.33) equals

n-1 k=0 1 2 -x - k n = 1 2 -nx, = ψ(nx).
This concludes the proof of the lemma.

Total decomposition sets

One of the main technical difficulties in this paper is to work out computations involving the greatest common divisors and the least common multiples of subsets of {r 1 , . . . , r } for r = (r 1 , . . . , r ) an -tuple of positive integers. The following lemma is particularly convenient to do so. A proof of it can be found in [19, page 112]. In particular, we have

r j = S j d(
H (Y ; r) = 1 0 β (mod lcm[r] 2 ) j=1 Y r 2 j + ψ Y -u -β r 2 j -ψ -u -β r 2 j du. 4.3.3 -Analysis of H (Y ; r) 103
Démonstration. Notice that the innermost product on the right-hand side above equals # (h 1 , . . . , h ) ∈ Z ; -u < h 1 , . . . , h ≤ Y -u and h j ≡ β (mod r 2 j ) for all j ∈ [ ] Hence, it suffices to prove that κ(r; h) = 1 if and only if there exists (a unique) β modulo lcm[r] 2 such that h j ≡ β (mod r 2 j ) for every j ∈ [ ]. One direction is easy. Indeed, if we suppose that such β exists, then for every i, j ∈ [ ], we have that h i ≡ h j ≡ β (mod gcd(r i , r j ) 2 ). Thus for every i, j ∈ [ ], gcd(r i , r j ) 2 | h i -h j . This proves that κ(r, h) = 1.

The opposite implication is more involved and uses the total decomposition set of r. Let {d(r; S)} be the total decomposition set of r. Since all the r j are squarefree, then the d(r, S) are pairwise coprime. Thus the conditions gcd(r i , r j ) 2 | h i -h j , for all i, j ∈ [ ] are equivalent to h i ≡ h j (mod d(r, S) 2 ), for every S ∈ P ≥1 ( ) and i, j ∈ S. This is equivalent to the existence, for every S ∈ P ≥1 ( ), of certain β S (mod d(r, S) 2 ) such that h j ≡ β S (mod d(r, S) 2 ), for every S ∈ P ≥1 ( ) and j ∈ S.

The Chinese remainder theorem now gives the result (recall that the d(r, S) are pairwise coprime).

The following Lemma should be compared to (4.38) and Lemma 4.3.4. This is the point of the proof where the cancellation of the secondary terms mentioned in section 4.1.3 take place. Lemma 4.3.5. Let C (Y ; q) be as in (4.25). Then for every Y ≥ 1, and positive integers and q, the following holds :

C (Y ; q) = ∞ r 1 =1 (r 1 ,q)=1 . . . ∞ r =1 (r ,q)=1 µ(r 1 ) . . . µ(r ) lcm[r] 2 E (Y ; r), (4.39)
where

E (Y ; r) = 1 0 β (mod lcm[r] 2 ) j=1 ψ Y -u -β r 2 j -ψ -u -β r 2 j du. (4.40)
Démonstration. Let C (Y ; q) be the right-hand side of (4.39). By inclusion-exclusion, the inner product on the right-hand side of (4.40) equals

T ⊂[ ]    j ∈T -Y r 2 j j∈T Y r 2 j + ψ Y -u -β r 2 j -ψ -u -β r 2 j    .

Preparatory results

By replacing it in (4.40) and interchanging the order of summation and integration, we see that

E (Y ; r) = T ⊂[ ] j ∈T -Y r 2 j ω T (r), (4.41)
where

ω T (r) := 1 0 β (mod lcm[r] 2 ) j∈T Y r 2 j + ψ Y -u -β r 2 j -ψ -u -β r 2 j du. (4.42)
Notice that the inner product on the right-hand side of the above equation only depends on the class of β (mod lcm[r| T ] 2 ), where for each

T = {i 1 , . . . , i #T }, with 1 ≤ i 1 < . . . < i #T ≤ , we define r| T := (r i 1 , . . . , r i #T ). (4.43) Hence ω T (r) : = lcm[r] 2 lcm[r| T ] 2 1 0 β (mod lcm[r| T ] 2 ) j∈T Y r 2 j + ψ Y -u -β r 2 j -ψ -u -β r 2 j du = lcm[r] 2 lcm[r| T ] 2 H #T (Y ; r| T ) , (4.44
E (Y ; r) = lcm[r] 2 T ⊂[ ]   j ∈T -Y r 2 j   H #T (Y ; r| T ) lcm[r| T ] 2 . (4.45)
Changing the order of summation, we deduce the inequality

C (Y ; q) = T ⊂[ ] Θ(T )Θ ([ ]\T ), (4.46)
where

Θ(T ) = ∞ r i 1 =1 (r i 1 ,q)=1 . . . ∞ r i #T =1 (r i #T ,q)=1 µ(r i 1 ) . . . µ(r i #T ) lcm[r| T ] 2 H #T (Y ; r| T ) = B #T (Y ; q), (4.47)
and

Θ ([ ]\T ) = (-Y ) -#T j ∈T     ∞ r j =1 (r j ,q)=1 µ(r j ) r 2 j     = (-A q Y ) -#T . ( 4
C (Y ; q) = j=0 j (-A q Y ) -j B j (Y ; q), = C (Y ; q).

A modification of H (Y ; r)

In the following we introduce a slight modification of H (Y ; r) (defined in (4.38)) which will be very important when estimating E (Y ; r) for large values of the r j (see Lemma 4.4.2 below). For positive integers and q, and for Y ≥ 1, we let

H * (Y ; r) = 1 0 . . . -u<h j ≤Y -u, ∀j∈[ ] h j distinct κ(r; h)du. (4.49)
The rest of this section is dedicated to the study of the interplay between H (Y ; r) and H * (Y ; r), specially when it comes to their contribution to C (Y ; q). Lemma 4.3.6. Let H * (Y ; r) be as in (4.49). Then we have the equality

H * (Y ; r) = lcm[r] 2 r 2 1 . . . r 2 Y + O( 2 (Y + 1) -1 ),
uniformly for every Y ≥ 1, every ≥ 1 and every r ∈ Z >0 .

Démonstration. We prove it by induction. For = 1, κ(r, h) = 1 for every integer h and every r ∈ Z >0 . Hence H * 1 (Y ; r) = Y + O(1). Suppose now that Lemma 4.3.6 is true for a certain ≥ 1. Let r = (r 1 , . . . , r +1 ) ∈ Z +1 >0 . Then, with the notation from the previous lemma, we have

H * +1 (Y ; r) = 1 0      . . . -u<h j ≤Y -u, ∀j∈[ ] h j distinct κ(r| [ ] ; h 1 , . . . , h ) -u<h +1 ≤Y -u h +1 ∈{h 1 ,...,h } ν(r, h)      du. (4.50) where ν(r, h) := 1, if (r j , r +1 ) 2 | h j -h +1 for all j = 1, 2, . . . , , 0, otherwise.
By the Chinese remainder theorem, whenever κ(r| [ ] , h 1 , . . . , h ) = 1, the inner sum equals (in this case, the congruences are always compatible)

Y gcd(r +1 , lcm[r| [ ] ]) 2 + O( ).
Replacing it in (4.50), we have the equality

H * +1 (Y ; r) = Y gcd(r +1 , lcm[r| [ ] ]) 2 H * Y ; r| [ ] + O( (Y + 1) ). since H * Y ; r| [ ] ≤ (Y + 1
H (Y ; r) = H * (Y ; r) + lcm[r] 2 r 2 1 . . . r 2 Y -1 1≤j 1 <j 2 ≤ gcd(r j 1 , r j 2 ) 2 + O( 4 (Y + 1) -2 ),
uniformly for every Y ≥ 1, every ≥ 2 and every r ∈ Z >0 .

Démonstration. The inner sum in (4.38) counts -tuples satisfying a certain congruence condition. We break up this counting in three terms. The first consisting of the -tuples containing exactly distinct elements, the second of those containing exactly -1 distinct elements, and the last one of those containing at most -2 distinct elements. The third one can be bounded by 4 (Y + 1) -2 . After integrating with respect to u, we obtain the equality

H (Y ; r) = H * (Y ; r) + 1 0 . . . -u<h j ≤Y -u, ∀j∈[ ] #{h 1 ,...,h }= -1 κ(r; h)du + O 4 (Y + 1) -2 . ( 4

.51)

Notice that the condition #{h 1 , . . . , h } = -1 means that exactly two of the h j that are equal and all the other are distinct and different from this common value.

For each j 1 , j 2 such that 1 ≤ j 1 < j 2 ≤ , let s j 1 ,j 2 be the ( -1)-tuple given by s j 1 ,j 2 = (r 1 , . . . , r j 1 , . . . , r j 2 , . . . , r , lcm[r j 1 , r j 2 ]) ,

where the means that these entries must be suppressed. Then the middle term on the righthand side in (4.51) equals Actually, for ≥ 3, the sum over T ⊂ [ ] in the above equation vanishes. Indeed, by changing the order of summation, this sum equals

1≤j 1 <j 2 ≤
gcd(r j 1 , r j 2 ) 2

T ⊂[ ] T j 1 ,j 2 (-1) -#T = 0, as a consequence of the binomial formula for the inner sum on the right-hand side above. We have just proved where the implied constant is absolute.

Estimating E (Y ; r)

In this section we give the two major lemmas that we use to estimate E (Y ; r). They will be applied, roughly speaking, to small and large values of the r j , respectively. , where e 1 denotes the multiplicative inverse of e 1 (mod r 2 1 ) and r 1 the multiplicative inverse of r 1 (mod e 2 1 ). An application of Lemma 4.3.1 now gives

S 1 (β) = ψ Y -u -β r 2 1 -ψ -u -β r 2 1
.

Summarizing, we have E (Y ; r) = E (Y ; r 1 , r 2 , . . . , r ).

Repeating the process for j = 2, . . . , and defining r 2 , . . . , r similarly, we obtain that E (Y ; r) = E (Y ; r), where r = ( r 1 , . . . , r ). The lemma now follows from applying (4.54) to the term on the right in the above equation.

Before stating our next lemma, we shall need some extra notation. These are given in the following definition. where the symbol means that the sum is restricted to the -tuples h such that h i = h j for all 1 ≤ i < j ≤ .

Note that κ(r(d), h) = 1 if and only if for every S ∈ P ≥2 ( ) and for every i, j ∈ S we have

d 2 S | h i -h j . Fix j 0 ∈ [ ].
At the cost of introducing some divisor functions, we can drop from the sum in the left-hand side of (4.58) the sum over the d S for which j 0 ∈ S, loosing at most a factor Y . Thus we have ; d S pairwise coprime, (d S , q) = 1 for all S ∈ P ≥1 ( ) .

T ∆ , Y + ∆ -2-2 .
Replacing it in (4.70), we obtain an upper bound for S >∆ : . This is the point when the quantity u p (h) (defined in (4.5)) makes its first appearance. Once again for simplificity, we write u p instead of u p (h). Notice that, for p coprime with q, the congruence ξ(n) ≡ 0 (mod p) has exactly u p solutions modulo p 2 . Therefore, by the Chinese remainder theorem, the congruence ξ(n) ≡ 0 (mod d)

S >∆ , Y + ∆ -2-2 + Y -2 ∆ -1+ . ( 4
has U d := p|d u p
solutions (mod d 2 ), for d squarefree, (d, q) = 1. A simple consequence of this remark is that N d (X 1 , X 2 , q; h) := ϕ(q) q U d d 2 (X 2 -X 1 ) + O(τ (q)U d ) (4.78) uniformly for 0 ≤ X 1 < X 2 and d squarefree, (d, q) = 1.

Now we break up (4.77) relatively to the possible values of d S(X 1 , X 2 , q; h) = S 1 + S 2 , (

where

                 S 1 = 1≤d≤y (d,q)=1
µ(d)N d (X 1 , X 2 , q; h),

S 2 = y<d≤X (d,q)=1 µ(d)N d (X 1 , X 2 , q; h), (4.80)
where X := max 1≤i≤j (X 2 + h i q) and y is a parameter to be chosen later depending on X.

For S 1 , a direct application of (4.78) gives

S 1 = 1≤d≤y µ(d) ϕ(q) q U d d 2 (X 2 -X 1 ) + O(τ (q)U d ) = ϕ(q) q (X 2 -X 1 ) d≥1 (d,q)=1 µ(d)U d d 2 + O   X d>y U d d 2 + d≤y τ (q)U d   = ϕ (
q) q A q (h)(X 2 -X 1 ) + O ,j (Xy -1+ + τ (q)y 1+ ), (4.81) where in the last line we use that U d = p|d u p ≤ j ω (d) .

For large values of d, formula (4.78) is not as meaningful. We look rather for an upper bound

N (D 1 , D 2 ) ≤ d 2 ∼D 2 g|d 2 2 v≤ X D 2 1 g d 1 ∼D 1 d 2 1 v≡w (mod z) 1 ≤ 2 d 2 ∼D 2 g|d 2 2 X D 2 1 g D 1 z + 1 τ (d 2 ) X d 2 ∼D 2 X D 2 1 D 1 d 2 2 + 1 X X D 1 D 2 + XD 2 D 2 1 .
As a consequence of (4.83), we deduce Finally, from (4.82), we have S 2 ,j X 1+ y -1 j . (4.87) By (4.81), (4.87) and (4.79) we obtain the equality S(X 1 , X 2 , q; h) = ϕ(q) q A q (h)(X 2 -X 1 ) + O Xy -1+ + τ (q)y 1+ + X 1+ y -1 j .

N (D 1 , . . . , D j ) ,j X X D 1 D 2 + XD 2 D 2 1 . ( 4
We make the choice y = X j j+1 in the equation above and establish the equality (4.75). This concludes the proof of Theorem 4.6.1.

1 1. 1

 11 Introduction I Répartition de fonctions arithmétiques . . . . . . . . . . . . . . . . . . . . . . . II Majorations en moyenne sur a modulo q . . . . . . . . . . . . . . . . . . . . . . III Une question de dépendance de variables aléatoires . . . . . . . . . . . . . . . . IV Moments d'ordre supérieur . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V Sommes d'exponentielles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Squarefree numbers in arithmetic progressions Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Proof of Theorem 1.1.3 assuming Theorem 1.1.4 . . . . . . . . . . . . . . . . . . 1.4 Initial Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Preparatory results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Main term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Bounding S >y [m](X, q) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Proof of Theorem 1.1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9 Exponential Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 On Bourgain's Bound and applications to squarefree numbers 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Initial Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Useful lemmata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

INTRODUCTION 11 Exemple 2 .

 112 Les fonctions nombre de diviseurs généralisées. On considère la fonction nombre de diviseurs classique donnée, pour tout n ≥ 1, pard(n) := #{d ≥ 1; d | n}.Plus généralement, on considère, pour k ≥ 2 les fonctions données, pour tout n ≥ 1, pard k (n) := #{(n 1 , . . . , n k ) ∈ Z k ≥1 ; n 1 . . . n k = n}.

9 20 , 5 12, 8 3k , pour k ≥ 7 .

 20587 pour k = 5, pour k = 6, On discutera du terme d'erreur pour d k en section II.

2 3 +

 3 ,(1.6) gives an asymptotic formula for q≤Q

  We start by studying S 2 . We haveS 2 (r; m 2 , q; b, c, d) = + dm 2 qα 2 , r),(1.100)whereδ(x, n) = n if n | x, 0 otherwise. If r | dm 2 , equation (1.100) becomes S 2 (r; m 2 , q; b, c, d) = , r) = δ(b, r)δ(c, r).So, S 2 (r, q; b, c, d) = 0 unless r divides both b and c, in which case, S 2 (r; m 2 , q; b, c, d) = r 2 = r(r, b, c, dm 2 ).

Lemma 2 . 4 . 1 .

 241 Let h(d) be as in (2.11) and let β be the multiplicative function defined by the formula h(d) = mn=d β(m), d ≥ 1.

  Theorem 1], [2, Proposition 1.4] or [35, Theorem 3]. The proof, for instance, follows the lines of [2, Proposition 1.4]. Lemma 2.4.4. Let X ≥ 1 and let , r be integers such that r is squarefree. Let I(X, , r) := u ∈ R; u and ru + ∈ (0, X) (2.23) and S( , r) := n∈I(X, ,r) µ 2 (n)µ 2 (rn + ).

  n∈I(X, ,r) d|ξ(n) µ(d) = d≥1 µ(d)N d ( , r), ON BOURGAIN'S BOUND AND APPLICATIONS TO SQUAREFREE NUMBERS 61 where N d ( , r) = # n ∈ I(X, , r); ξ(n) ≡ 0 (mod d) .

  , , r)| + O r X log y y + y log y . (2.32) For large values of d, formula (2.30) is useless. Instead of it we will deduce by different means an estimation for N >y ( , r) := d>y µ(d)N d ( , r)

  79

Démonstration. 2 ,

 2 The proof follows exactly as that of [17, Theorem 7.1], taking into account the simple inequality n (mod p) ∈Ωp for every p|P for any real numbers ρ d supported on d | P with ρ 1 = 1.

  .41) Adding together (3.15), (3.33) and (3.41), we have Proofs of the results

  |A d | = α (mod d) α (mod p)∈Ωp for every p|d G (d, α) , (3.57) where G (d, α) := n≡α (mod d) a n . (3.58) We use the ψ function to evaluate G (d, α). Injecting (3.56) in (3.58) and interchanging the order of summation, gives

  1.5 now follows by combining (3.13), (3.14) and (3.64).

2 IntroductionCorollary 4 . 1 . 4 .

 2414 For every > 0, the inequalities

1

 1

4. 1 . 2 -

 12 Arithmetic progressions 97 (4.20) gives square root cancellation relative to the main term. Montgomery conjectured (see [7, top of the page 145]) that something even stronger should hold : E(X, q, a) = O (X/q)

  use the classical notations a ∼ A for A/2 < a ≤ A ; τ k (n) are the generalized divisor functions and count the number of k-tuples of positive integers (d 1 , . . . , d k ) such that n = d 1 . . . d k , τ (n) := τ 2 (n) is the classical divisor function ; -We write ω(n) to denote the number of its prime divisors ; -Z >0 denotes the set of positive integers, and [ ] := {1, . . . , } ; -If r = (r 1 , . . . , r ) is a -tuple of positive integers, we denote lcm[r] := lcm[r 1 , . . . , r ]; and gcd[r] := gcd[r 1 , . . . , r ];-If A and B are two sets, we writeB A = {x = (x α ) α∈A ; x α ∈ B ∀α ∈ A};-For a finite set S, #S denotes its cardinality and for an interval I ⊂ R, |I| denotes its length.-For a finite set T and i a positive integer, we use the abbreviations P ≥i (T ) = {S ⊂ T ; #S ≥ i}. Furthermore we write P ≥i ( ) = P ≥i ([ ]) ; -Unless otherwise stated, the implied constant by the symbols O and depend at most on the subscribed variables (e.g. the implied constant by the symbol , depends at most on and ) ; -The ψ function is the classical sawtooth function defined by ψ(x) = x -

Lemma 4 . 3 . 2 .

 432 For each r ∈ Z >0 , there exists a unique (2 -1)-tuple of integers (d(r, S)) S∈P ≥1 ( ) , called the total decomposition set of r, such that for every T ∈ P ≥1 ( ), we have gcd(r j ; j ∈ T ) = S⊃T d(r; S), (4.34) and lcm[r j ; j ∈ T ] = S∩T =∅ d(r; S). (4.35)

Proposition 4 . 3 . 8 .

 438 Let ≥ 3 be an integer. Let Y ≥ 1. Let r ∈ Z >0 , where all the r j are squarefree. Let E (Y ; r) be as in (4.40) and E * (Y ; r) as in (4.53). Then we have the equality E (Y ; r) = E * (Y ; r) + O 2 4 (Y + 1) -2 ,

Lemma 4 . 4 . 1 . 1 0

 4411 Let ≥ 1. Let r ∈ Z >0 , where all the r j are squarefree. Let {d(r, S)} be the total decomposition set of r as in Lemma 4.3.2. Then we have the equality|E (Y ; r)| ≤ S∈P ≥2 ( ) d(r, S)2 , where E (Y ; r) is as in (4.40). Démonstration. From (4.40) we deduce the inequality |E (Y ; r)| ≤ lcm[r] 2 , (4.54) which is larger than the bound claimed in Lemma 4.4.1 by a factor of j∈[ ] d(r, {j}) 2 (see (4.35)). We show how to recover the factor d(r, {1}) 2 and the same process can be applied for d(r, {j}), j ≥ 2. Estimating E (Y ; r) Let e 1 := d(r, {1}) and r 1 be such that r 1 = e 1 r 1 . Then, by the Chinese remainder theorem, (4.40) can be written as E (Y ; r) = β (mod lcm[ r 1 ,r 2 ,...,r ] 2 )

Definition 4 . 4 . 1 .

 441 Let ≥ 1. Let D = (D S ) S∈P ≥2 ( ) be a (2 --1)-tuple of positive real numbers. Let e be an -tuple of pairwise coprime squarefree integers. Let X (D, e) denote the set of (2 -1)-tuples of positive integers d = (d S ) S∈P ≥1 ( ) such that the d S are pairwise pairwise coprime, d S ∼ D S for all S ∈ P ≥2 ( ), and d {j} = e j for all j ∈ [ ]. For every (2 -1)-tuple of pairwise coprime squarefree integers d = (d S ) S∈P ≥1 ( ) , for every j ∈ [ ], we associate the integers r j (d) := the -tuple r(d) := (r 1 (d), . . . , r (d)).

Finally, for every Y ≥ 1 ,

 1 define the sum of error terms S (Y ; D, e) := . . . d∈X (D,e) |E * (Y ; r(d))| , (4.56) where E * (Y ; D, e) is as defined in (4.53).

4. 3 . 4 -

 34 A modification of H (Y ; r) 109 Remark that the -tuple r(d) given by definition (4.55) is the unique -tuple whose total decomposition set (see Lemma 4.3.2) satisfies d(r(d), S) = d S for every S ∈ P ≥1 ( ).

Lemma 4 . 4 . 2 .-

 442 Let > 0 and ≥ 2 be an integer. Then, we have the inequality S (Y ; D, e) uniformly for Y ≥ 1, every (2 --1)-tuple of positive real numbers D = (D S ) S∈P ≥2 ( ) and every -tuple of positive integers e = (e 1 , . . . , e ), where S(Y ; D, e) is as in Definition 4.4.1 above. Démonstration. We first prove the closely related inequality . . . d∈X (D,e) H * (Y ; r(d)) u<h j ≤Y -u ∀j∈[ ] κ(r(d), h)du,

(d S )∈X ,j 0 (D) 1 0.

 1 -u<h j 0 ≤Y -u j =j 0 where X ,j 0 (D) is a slight modification of X (D, e) given by X ,j 0 (D) = (d S ) S∈P ≥2 ( ), j 0 ∈S ; d S pairwise coprime, d S ∼ D S for every S such that j 0 ∈ S ∈ P ≥2 ( ) , (4.60) Injecting it in (4.59), we obtain that . . . d∈X (D,e) H * (Y ; r(d)) 4.61) holds for arbitrary j 0 ∈ [ ]. Thus by taking the minimum over j 0 ∈ [ ], we see that the left hand side of (4.61) is exponent for the D S occurs when #S = 2. This concludes the proof of (4.58). With a very similar argument, we can prove that for every T ⊂ [ ], we have the inequality T = {i 1 , . . . , i #T } with 1 ≤ i 1 < . . . < i #T ≤ , r(d)| T is given by (compare with (4.43)) r(d)| T = r i 1 (d), . . . , r i #T (d) .

4. 3 . 4 -.

 34 A modification of H (Y ; r) 111 If we replace E * (Y ; r) by its definition (4.53) in (4.56), we obtain the inequality S (Y ; D, e) ≤ . . .d∈X (D,e) T ⊂[ ] lcm[r(d)] 2 lcm[r(d)| T ] 2 j ∈T Y r j (d) 2 H * #T (Y ; r(d)| T ) . . . d∈X (D,e) T ⊂[ ] Y ; r(d)| T ) , (4.64)where in the second line we used (4.55) and the fact that for every d ∈ X (D, e), the d S are pairwise coprime and that for every S ∈ P ≥2 ( ), we have d S ∼ D S .By changing the order of summation in (4.64) and using (4.63), we obtain the boundS (Y ; D, e) , Y + T ⊂[ ] Φ T (D), -3 if #S ≥ 2, #S ∩ T ≤ 1, 2#(S\T ) + 2/ if #S ≥ 2, #S ∩ T ≥ 2.By (4.66) and ≥ 2, we obtainΦ T (D)By injecting it in (4.65), we complete the proof of Lemma 4.4.2.

  r ,q)=1 µ(r 1 ) . . . µ(r ) lcm[r] 2 E (Y ; r). Using total decomposition sets (see Lemma 4.3.2 and Remark 4.3.1 following it), we can write it

5 . 2 -=

 52 and r(d) is as in (4.55). For every d ∈ Z P ≥1 ( ) >0 , we define N 2 (d) := S∈P ≥2 ( ) d S . We break up the sum in the righthand side of (4.67) according to the size of N 2 (d). Let ∆ ≤ Y 2 to be chosen optimally later. Then we have C (Y ; q) the second line, we made the change of variables n = N 2 (d), and e j = d {j} . 4.5.2 Study of S >∆ By Proposition 4.3.8, we have 4.Study of S >∆ 113 T ∆ + O , Y -2 ∆ -1+ , (4.70) say.Note that whenever E * (Y ; r(d)) = 0, then for every S ∈ P ≥2 ( ), there are distinct h, h such thath, h ∈ (-u, Y -u] and d 2 S | h -h . Hence d S < √ Y ,for every S ∈ P ≥2 ( ). By dyadic decomposition, there exists D = (D S ) S∈P ≥2 ( ) , such that 1 ≤ D S ≤ √ Y for every S ∈ P ≥2 ( ), satisfying S∈P ≥2 ( ) . e 2 S(Y ; D, e), (4.71) where S(Y ; D, e) is as in (4.56), where > 0 is arbitrary. By Lemma 4.4.2, we deduce from (4.71)

2 3

 2 .72) Gathering (4.68), (4.69) and (4.72), we deduce the inequalityC (Y ; q) , ∆ 1+ + Y + ∆ -2-2 + Y -2 ∆ -1+ . (4.73) Theorem 4.1.6 now follows from (4.73) and the optimal choice ∆ = Y max +2 , -2 2

1 j and D 1 D 2 > y 2 j

 1122 .85) By a completely similar argument, we have the same result for any pair (D i , D j ) instead of (D 1 , D 2 ). Thus we can supposeD 1 ≥ D 2 ≥ D i , for all i ≥ 3.(4.86) The inequality D 1 • • • D j > y and (4.86) imply . Therefore (4.85) implies N (D 1 , . . . , D j ) ,j X 1+ y -1 j .

4. 7

 7 Proofs of Corollaries 4.1.2 and 4.1.7 Let 0 < k ≤ x, k integer. We develop M (x, k) (see (4.1)) as follows M (x; k) =

0≤h 1 ,

 1 ...,h j <k S(0, x, 1; h).(4.88)

  .107) Since t ≤ s holds, equation (1.104) tells us that we only need to prove that t ≤ s -u. That is

	s -t 2	≤ s -t,
	which holds since s -t ≥ 0. Thus, (1.107) is true. This completes the proof of (1.98) and (1.99)
	At last, for (1.95), (1.96), we write

  to retrieve Hall's upper bound for C (k) (see (4.12) above). The upper bound in Lemma 4.4.2 seems far from optimal, but improving it would only have an effect on Theorem 4.1.6 for = 3 or 4, since for larger values the other error term dominates, and we are not certain that one coumd get a much better error term in Proposition 4.3.8. In a different direction, when = 2, one can deduce from (4.40) and Lemma 4.3.1 that

	100	Notation
	which would suffice
		2 concerns the non-diagonal
	terms. Proposition 4.3.8 and Lemma 4.4.2 can be replaced by a very slight modification of
	Lemma 4.3.6 giving
	E (Y ; r)	Y -1 ,

  Let n be a positive integer and x a real number. Let ψ be as defined in (4.32).

	4.3.1 -Total decomposition sets				101
	Lemma 4.3.1. Then we have the equality
	n-1 k=0	ψ x +	k n	= ψ(nx).	(4.33)

  r; S). In these notes we only use Lemma 4.3.2 in the case where all the r j are squarefree. Following Hall's proof, we see that in this case the d(r, S) take the form Lemma 4.3.4. Let ≥ 1 and r = (r 1 , . . . , r ) ∈ Z >0 , where all the r j are squarefree. Let H (Y ; r) be as in (4.38) with Y ≥ 1. Then we have the equality

		µ(r 1 ) . . . µ(r ) lcm[r] 2	H (Y ; r),	(4.37)
	where r = (r 1 , . . . , r ) and
	1
	H (Y ; r) :=	. . .	κ(r; h) du,	(4.38)
	0	-u<h j ≤Y -u,
		∀j∈[ ]
	with κ(r; h) as in (4.36).
	4.3.3 Analysis of H (Y ; r)
	Remark 4.3.1. p.

d(r, S) := p|gcd(r j ;j∈S) p|lcm[r j ;j ∈S]

  ) . The result for + 1 is now a consequence of the induction hypothesis. This concludes the proof of the Lemma. Lemma 4.3.7. Let H (Y ; r) be as in (4.38) and H * (Y ; r) as in (4.49). Then we have the equality

  lcm[r j 1 , r j 2 ] 2 + O( 4 (Y + 1) -2 ), (4.52) by Lemma 4.3.6. Finally, Lemma 4.3.7 now follows from (4.51) and (4.52).4.3.4 -A modification of H (Y ; r)1≤j 1 <j 2 ≤ j 1 ,j 2 ∈Tgcd(r j 1 , r j 2 ) 2 +O 2 4 (Y + 1) -2 .

						107
	Inspired by (4.45), we define
	E * (Y ; r) := lcm[r] 2	T ⊂[ ] j ∈T	-Y r 2 j	H * #T (Y ; r| T ) lcm[r| T ] 2 .	(4.53)
	By Lemma 4.3.7 we have the equality
	E (Y ; r) = E * (Y ; r)+	lcm[r 2 ] r 2 1 . . . r 2 Y -1	T ⊂[ ]	(-1) -#T
		lcm[r] 2 r 2 1 . . . r 2 Y -1	1≤j 1 <j 2 ≤	r 2 j 1 r 2 j 2

1≤j 1 <j 2 ≤ H * -1 (Y ; s j 1 ,j 2 ) =

Remerciements

Proof of the main Theorem

Proof of the main Theorem

We start by recalling (2.15) C[γ r,s ](X, q) = S[γ r,s ](X, q) -2C(q) X q

By Proposition 2.5.1 and formula (2.17), we deduce the inequality C[γ](X, q) ,r q 1+ + X 1/2 q 1/2 (log q) -δ +

The proof of Theorem 2.1.2 is now complete.

Chapitre 3

On two conjectures concerning squarefree numbers in arithmetic progressions

Introduction

The distribution of arithmetic sequences in arithmetic progressions is a central subject in analytic number theory. Let f : Z >0 → R >0 be a positive arithmetic sequence. If f is sufficiently reasonable, one expects that, for all (a, q) = 1, we have

Two important questions that arise naturally concern the uniformity of such a formula (see section 3.1.1 below) and the size of the error term

These questions are intimately related to the analytic properties of the associated L-functions

where χ is a Dirichlet character. Two particularly interesting cases occur when f = Λ is the van Mangoldt function or f = µ 2 , where µ is the Möbius function. The first one is closely related to the distribution of prime numbers while the latter corresponds to squarefree numbers. For such choices the associated L-functions are, respectively,

Preparatory results

We also remark that (4.35) implies that the r i are squarefree if and only if the d(r, S) are squarefree and pairwise coprime.

Unfolding A q (h)

We point out that it is not very practical to work with A q (h) under the form of an infinite product. Thus we unfold this infinite product in order to obtain a simpler expression. More precisely, we have the following Lemma 4.3.3. Let h = (h 1 , . . . , h ) ∈ Z , and let A q (h) be as in (4.27). Then we have the equality

where r = (r 1 , . . . , r ), and

Démonstration. A proof in the case q = 1 can be found in [18, page 16], and the general case follows in a completely analogous way.

Using Lemma 4.3.3in the defnition 4.26, we deduce that

Counting tuples of squarefree numbers

Counting tuples of squarefree numbers

In this section we study the following generalization of S(x; h) (see

for an j-tuple of integers h = (h 1 , . . . , h j ) and a positive integer q such that X 1 + h i q ≥ 0, i = 1, . . . , j. Sums of this type were vastly studied in the past, although, in general, the condition (n, q) = 1 was not taken into account. For example, we mention the articles by Mirsky [START_REF] Mirsky | Arithmetical pattern problems relating to divisibility by rth powers[END_REF],

Tsang [START_REF] Tsang | The distribution of r-tuples of square-free numbers[END_REF] and, more recently, Reuss [START_REF] Reuss | Pairs of k-free numbers, consecutive square-full numbers[END_REF]. The result from [START_REF] Reuss | Pairs of k-free numbers, consecutive square-full numbers[END_REF] allows to obtain quite small error terms but not assuring uniformity relatively to max(h 1 q, . . . , h j q), which is essential here. We will content ourselves with a more modest result (with a larger error term), but that will be enough for our purposes. The main theorem of this section is the following Theorem 4.6.1. For every > 0, for every j ≥ 1, we have the equality

for every X 1 , X 2 such that 0 ≤ X 1 < X 2 , every j-tuple of integers h = (h 1 , . . . , h j ), and every positive integer q such that q ≥ 1 such that X 1 + h i q ≥ 0, i = 1, . . . , j, where S(X 1 , X 2 , q; h) is as in (4.74) and A q (h) is as in (4.27).

Remark 4.6.1. A better error term at this point would reflect in a larger value for δ (see (4.17)), but would not help to improve the exponent on Theorem 4.1.1.

Démonstration. We start by defining, as in [START_REF] Reuss | Pairs of k-free numbers, consecutive square-full numbers[END_REF] and [START_REF] Tsang | The distribution of r-tuples of square-free numbers[END_REF], some functions related to squarefree numbers and the sum we are interested in. Let σ(n) :=

Notice that the right-hand side of equation (4.76) above actually depends on h and q. But since these numbers will be held fixed in the following calculation, we omit this dependency. Since

where

(n, q) = 1 and ξ(n) ≡ 0 (mod d)}.

Remark that the decomposition above is not in general unique. Furthermore since d is squarefree, the d i are pairwise coprime. By dyadic decomposition, one can find

where

We need to give an upper bound for N (D 1 , . . . , D j ). We start by noticing that we can forget about the variables (d i , u i ), for i ≥ 3 at the cost of introducing some divisor function. That is

where N (D 1 , D 2 ) is defined similarly to N (D 1 , . . . , D j ). It is clear from the formula above that any u 1 counted above must satisfy

Thus we have the inequality

We write

and the congruence

Therefore we have

-Study of S >∆

where h = (h 1 , . . . , h j ) and S(0, x, 1; h) is as in (4.74). It follows from Theorem 4.6.1 that the sum over the h 1 , . . . , h j equals (cf. definition (4.8))

By formula (4.88), we obtain the equality

where C (k) is defined in (4.7). Corollary 4.1.2 is now a simple consequence of Theorem 4.1.1 and equation (4.89) above.

Corollary 4.1.7 follows in a much similar fashion. We start, as before, by expanding M(X, q; ) (see (4.22)). We obtain the formula

We make the change of variables n j = n and n i = n j + f i q, for i = 1, . . . , j -1. Thus, we are allowed to write S j (X; q) = . . .

where for every j-tuple of integers h = (h 1 , . . . , h j ), we write

Note that whenever I(X, q; h) = ∅, we have I(X, q; h) = (X 1 , X 2 ], where X 1 , X 2 are real numbers satisfying 0 ≤ X 1 + h i q < X 2 + h i q ≤ X, i = 1, . . . , j.

Hence, we may use Theorem 4.6.1 for the inner sum on the right-hand side of (4.91). After summing over f 1 , . . . , f j-1 , we see that S j (X; q) = . . . where, in the second line, we observed that whenever |f i | > X q for some 1 ≤ i ≤ j -1, then |I(X, q; f , 0)| = 0.

In what follows next, we evaluate the sum over the f i above. To this purpose we have the following : Lemma 4.7.1. With the above notation, and B j as defined in (4.26), we have for every X > 0 and every integer q, the inequality

Démonstration. We first notice that

where for each measurable set A ⊂ R, χ A denotes its characteristic function and in the second line we made the change of variables v = qu + qf , where u ∈ (0, 1] and f ∈ Z. Summing over f 1 , . . . , f j-1 and making the change of variables

we obtain, since A q (f , 0) = A q (h) (recall (4.27)),

A q (f , 0) |I(X, q; f , 0)| = q h∈Z j A q (h)

Finally, by interchanging the order of summation and integration, we see that

A q (f , 0) |I(X, q; f , 0)| = q 1 0 . . .

-u<h 1 ,...,h j ≤Y -u A q (h)du, which concludes the proof of Lemma 4.7.1. Now, Lemma 4.7.1 when applied in (4.92) gives the equality S j (X; q) = ϕ(q)B j X q ; q + O , X M(X, q; ) = ϕ(q)C X q ; q + O , X +1 + X q