Olivier Ramaré
email: olivier.ramare@univ-amu.fr

G K Viswanadham
email: viswanadh@iiserbpr.ac.in

Modular Ternary Additive Problems with Irregular or Prime Numbers

Our initial problem is to represent classes m modulo q by a sum of three terms, two being taken from rather small sets A and B and the third one having an odd number of prime factors (the so-called irregular numbers in S. Ramanujan's terminology) and lying in an interval [q 20r , q 20r + q 16r ] for some given r ≥ 1. We show that it is always possible to do so provided that |A| • |B| ≥ q(log q) 2 . The proof leads us to study the trigonometric polynomials over irregular numbers in a short interval and to seek very sharp bounds for them. We prove in particular that q 20r ≤s≤q 20r +q 16r e(sa/q) q 16r (log q)/ ϕ(q) uniformly in r, where s ranges over the irregular numbers. We develop a technique initiated by Selberg and Motohashi to do so. In short, we express the characteristic function of the irregular numbers via a family of bilinear decompositions akin to Iwaniec's amplification process, and that uses pseudo-characters or local models. The technique applies to the Liouville function, to the Möbius function and also to the von Mangoldt function, in which case it is slightly more difficult. It is however simple enough to warrant explicit estimates, and we prove, for instance, that X< ≤2X Λ( ) e( a/q) ≤ 1300 √ q X/ϕ(q) for 250 ≤ q ≤ X 1/24 and any a prime to q. Several other results are also proved.

INTRODUCTION

We investigate the additive behavior of irregular numbers in Z/qZ when q is large. We borrow the terminology irregular numbers from Ramanujan in [START_REF] Ramanujan | Irregular numbers[END_REF]: they are the integers having an odd number of prime factors (see sequence A028260 in [START_REF]The On-Line Encyclopedia of Integer Sequences[END_REF]). The importance of such numbers increased sizeably when Selberg enunciated his parity principle around 1949 (for instance, in [START_REF] Selberg | On elementary methods in prime number theory and their limitations[END_REF]), as such numbers behaved very similarly to usual integers from a sieve viewpoint. Their characteristic function is easily expressed in terms of more common players: it is (1λ(n))/2, where λ(n) = (-1) Ω(n) is the Liouville function, Ω(n) denoting as usual in such surroundings the number of prime factors of n counted with multiplicity.

Here is our main result.

Theorem 1. Let q ≥ 3 be a prime number and r ≥ 1 be a given number. Let S be the set of irregular numbers in [q 20r , q 20r + q 16r ]. Let A and B be two arbitrary sets in Z/qZ such that |A| • |B| ≥ q(log q) 2 . We have a+b+s≡m [q] a∈A, b∈B, s∈S [START_REF] Barban | On an extremal problem[END_REF]

∼ |A| • |B| • |S| q

(as q goes to infinity) valid for every m ∈ Z/qZ and uniformly in r.

The difficulty here lies in handling the exponential sum over irregular numbers, or equivalently, the exponential sum n λ(n)e(na/q) when q can be as large as a power of X and n runs over the short interval [X, X + X θ ]. This exponential sum is very similar (and can be reduced, by using the identity λ = μ ½ X 2 , where denotes the arithmetic convolution product: (f g)(n) = d|n f (d)g(n/d)) to the sum n μ(n)e(na/q). This latter sum is often compared to p e(pa/q) where p ranges over primes, but the short interval condition leads to difficulties. However, the method we develop is valid equally for λ(n), μ(n) or the characteristic function of the primes. The proofs are even somewhat easier in the cases of λ(n) and μ(n).

Historically, Davenport saw immediately the strength of Vinogradov's masterpiece [START_REF] Vinogradov | Representation of an odd number as a sum of three primes[END_REF] and used it for the trigonometric polynomial with Möbius coefficients in [START_REF] Davenport | On some infinite series involving arithmetical functions. II[END_REF]. His method is indirect, as he uses Vinogradov's estimate on the primes. The initial treatment of Vinogradov was involved; the book [START_REF] Vinogradov | The Method of Trigonometrical Sums in the Theory of Numbers[END_REF] gives an excellent account of it. Since then, the theory of bilinear decompositions for the primes has evolved with the systematic use of combinatorial identities, though mostly over the primes. It is folklore knowledge that such methods adapt mutatis mutandis to the case of the Möbius function, though finding references is not so easy. We found [4, Theorem 3], [START_REF] Hajela | On the maximum of an exponential sum of the Möbius function[END_REF]Theorem 2.1] and [START_REF] Iwaniec | Analytic Number Theory[END_REF]Theorem 13.9] as well as [40, Ch. II, § 6]. There has been a renewal of interest in this problem since then, and we mention in particular [START_REF] Bourgain | Disjointness of Moebius from horocycle flows[END_REF] and [START_REF] Green | The Möbius function is strongly orthogonal to nilsequences[END_REF].

Theorem 1 will be a consequence of the next estimate. Theorem 2. When q ≤ X η for some η < 1/8, T ≤ (X η /q) 16/13 and a is prime to q, we have

T -T ≤X μ(
) it e a q dt X log min(q, 2 + T ) ϕ(q) .

Variants of the theorem are also available:

(V2) One can replace μ( ) by λ( ) and get the same bound. (V3) One can add the coprimality condition ( , q) = 1 and replace 1/ ϕ(q) by 1/ √ q.

(V4) One can replace the Möbius function μ( ) by the von Mangoldt function Λ( ) and change 1/ ϕ(q) to √ q/ϕ(q).

Variant (V1) is the one of the theorem. All the results below will have the same variants. This L 1 -estimate readily leads to an estimate of the trigonometric polynomial over a short interval.

Theorem 3. Let η < 1/8 be given. When q ≤ X η , a is prime to q and θ 0 is defined by X 1-θ 0 √ q = (X η /q) 16/13 , we have

X< ≤X+X θ μ( ) e a q
X θ log q ϕ(q) for any θ ∈ (θ 0 , 1].

Comparing our bound to the one obtained by Zhan in [41, Theorem 2], we can say that the author has access to smaller intervals, but our bound has no power of log X. This may not be obvious to compare the two results, but a closer scrutiny shows that the constant B in [41, Theorem 2] is smaller than c 2 /2 -1.

The estimates above require that we stay at the precise point a/q and do not shift (except for some trivial amount) from it. When θ = 1, we can however relax this condition, and this may be important for the circle method (a summation by parts enables one to extend the result to |αa/q| 1/X; going further than that is the difficulty).

Here is what we get. Theorem 4. When q ≤ X η for some η < 2/13, a is prime to q, |β|X + |t| ≤ (X η /q) 13/2 and |β|X ≤ X (2-13η)/44 , we have ≤X μ( ) it e( β) e a q X ϕ(q) .

Here and thereafter, we do not try to get the best exponents, our aim being to describe precisely the method. These upper bounds have two features: after division by X θ (the trivial estimate), they tend to 0 with q (no power of log X comes in to weaken the result); and they are valid for q up to some power of X. The technique developed is to represent the Möbius function by a family of linear combinations of linear and bilinear sums in modern terminology (or of type I and type II sums if we are to follow Vinogradov's initial choice of words). Let us specify here that the results obtained are all effective (i.e., all implied constants can be explicitly determined); the possible Siegel zero thus limits our expectation as to what kind of results we may reach (it cannot however be properly termed an obstruction!).

A similar question over the primes has received attention. Theorem 2b of [START_REF] Vinogradov | The Method of Trigonometrical Sums in the Theory of Numbers[END_REF]Ch. IX] gives a first answer in that case (on selecting there ε = 2(log q)/log log X). A simplified version reads n≤X Λ(n) e na q X(log q) 10 √ q , q ≤ exp log log X 10 .

(1.1)

In [START_REF] Daboussi | Brun's fundamental lemma and exponential sums over primes[END_REF] the author obtained an estimate analogous to (1.1) but better when q does not have too many divisors. Both these results rely on bilinear decompositions and are thus extendable to the Möbius function. A first best possible result in the case of primes was reached in [START_REF] Ramaré | On Bombieri's asymptotic sieve[END_REF]: the function of q is √ q/ϕ(q). The variable q is still restricted to being not more than exp((log X) 1/3 /50), and, though flexible, the method developed also relies on positivity and is thus not adaptable to the case of the Möbius function. A. Karatsuba in [14, Ch. 10, Sect. 4, Lemma 7] gives the same coefficient √ q/ϕ(q) for q up to exp(c(log X) 1/2 ) for some positive constant c > 0, but this time by using analytical methods that are not transposable (more on this issue later). In fact, on using Gallagher's prime number theorem, i.e., [8, Eq. ( 5)], one can reach a result of similar strength: the same coefficient and for q up to some power of X. Our present method gives an analogous result without using log-free zero density theorem or even mentioning the zeros, and, in this aspect, this project is a sequel of Motohashi's work (see [START_REF] Motohashi | On Gallagher's prime number theorem[END_REF][START_REF] Motohashi | Primes in arithmetic progressions[END_REF]). Indeed, Motohashi in [START_REF] Motohashi | Primes in arithmetic progressions[END_REF] produced a proof of Gallagher's prime number theorem without using the zeros of the relevant Dirichlet L-series. Such a proof can be adapted to the case of the Möbius function by using the material we develop here. Let us record the result.

Theorem 5. There exist constants

c 1 , c 2 > 0 such that q≤Q * χ mod * q n≤X μ(n)χ(n) X exp -c 1 log X log Q provided that exp( √ log X) ≤ Q ≤ X c 2 .
The constant c 1 is chosen so that at most one Dirichlet L-function has a zero in the region σ ≥ 1c 1 /log(Q(2 + |t|)). Such a zero, if it exists, is simple and attached to a real character. The symbol * means that, in case such an exceptional zero exists, say β attached to the character χ * , we replace

n≤X μ(n)χ * (n) by n≤X μ(n)χ * (n) - X β βL (β, χ * ) . (1.2)
In case L (β, χ * ) is small, this contradicts Theorem 4. Indeed, on choosing X to be a large power of q = Q, we deduce from Theorem 5 that ≤X, ( ,q)=1 μ( ) e( β) e a q X √ q ϕ(q)L (β, χ * ) .

Let us record this result with the proper quantifiers.

Corollary 6. There exists a constant c > 0 such that if L(s, χ * ) admits a real zero β > 1c/log q, then L (β, χ * ) q/ϕ(q). This completes the result of Pintz [START_REF] Pintz | Elementary methods in the theory of L-functions. II: On the greatest real zero of a real L-function[END_REF], namely that, under the above assumptions, one has L (1, χ * ) q/ϕ(q). It is possible to go from one to the other when 1β = o((log q) -2 ) by using simple analysis.

The methods we use are fully explicit and even lead to possible numerical bounds. Because the most difficult case is the one of the primes, we consider this case and prove the following.

Theorem 7. When 250 ≤ q ≤ X1/24 and a is prime to q, we have

X< ≤2X Λ( ) e a q ≤ 1300 √ q X ϕ(q) .
It is worthwhile noting that the constant 1300 is maybe large but explicit, while the work [START_REF] Ramaré | On Bombieri's asymptotic sieve[END_REF] relies on a Brun sieve-based preliminary sieving process that would make such a computation very hard (it would also most probably result in a much higher constant). We made the effort to get an explicit constant, but there are many places where this work can numerically be improved upon. Notice that Platt in [START_REF] Platt | Numerical computations concerning the GRH[END_REF] has computed the zeroes of all the Dirichlet L-series of conductor q ≤ 4 • 10 5 whose imaginary parts are not more than 10 8 /q, so the distribution of primes up to such levels is better handled by using this.

Concerning the method, as already mentioned, we detect the primes via a family of bilinear decompositions, in a mechanism akin to Iwaniec's amplification process. Our implementation is inherited from a technique introduced by Selberg around 1973 (as mentioned by Bombieri in [START_REF] Bombieri | Le grand crible dans la théorie analytique des nombres[END_REF] and by Motohashi in [START_REF] Motohashi | Primes in arithmetic progressions[END_REF]), which has shown exceptional power.

This family of decompositions is shown to be "orthogonal," in some sense, via a large sieve extension of the classical large sieve inequality for the Farey points, that encompasses generalized characters in Selberg's or Motohashi's terminology, also called (up to some rescaling) local models in [START_REF] Ramaré | Arithmetical Aspects of the Large Sieve Inequality, with the collaboration of D. S. Ramana (Hindustan Book Agency[END_REF]. See Theorem 10 below, where the implied constant is improved with respect to the classical one.

The building of our decompositions involves Barban-Vehov 1 weights (see [START_REF] Barban | On an extremal problem[END_REF]). While studying an optimization problem close to the one that classically found the Selberg sieve for primes, Barban and Vehov introduced the weights

λ (1) d = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ μ(d) when d ≤ z, μ(d) log(z 2 /d) log z when z < d ≤ z 2 , 0 when z 2 < d. (1.3)
They consider in fact slightly more general weights with a y instead of z 2 (see Motohashi [19,Sect. 1.3] and Graham [START_REF] Graham | An asymptotic estimate related to Selberg's sieve[END_REF]). Their particular property is that 4 or not. We will follow an idea of Motohashi that the weaker property

n≤B d|n λ (1) d 2 B log z whether B ≥ z
n≥1 1 n 1+ε d|n λ (1) d 2 z ε (ε log z) 2 , ε∈ (0, 1],
can often be enough (this is our case) via Rankin's trick (see [START_REF] Rankin | The difference between consecutive prime numbers[END_REF]) and is easier to establish. The required material is contained in Lemma 27 and comes from [START_REF] Betah | Explicit expression of a Barban & Vehov theorem[END_REF]. From an explicit viewpoint, this implies dealing with sums of type d≤D μ(d)/d 1+ε or d≤D μ(d) log(D/d)/d 1+ε with some coprimality conditions added, and such sums are really more difficult to handle than the ones with ε = 0. In this latter case, identities can be used to effect that do not have any counterpart (as far as we can see) in the case ε > 0. Notation. Our notation is standard except maybe for f (x) = O * (g(x)), which means that |f (x)| ≤ g(x). Furthermore, the arithmetic convolution is denoted by ; it is defined by (f g)(n) = d|n f (d)g(n/d) under obvious conditions. We however require quite a lot of partial definitions to make the writing easier. It is then easy to forget the meaning of a quantity, and we try to recall the most important ones here. The letters θ and H are used in several acceptions.

Let us recall the definition of the Ramanujan sum as well as its evaluation in terms of the Möbius function: By ∼ L, we mean L < ≤ 2L. The main actor is

S a q , t, β = ∼X, ( ,q)=1 Λ( ) it e(β ) e a q , (1.6)
which we will split into a linear combination of sums of the linear type

L (1)
r (a, t, β) defined in (2.11) and L

(2) r (a, t, β) defined in (2.12), and of sums of bilinear type:

S r a q , t, β, M, N = mn∼X, (mn,q)=1 m∼M, n∼N Λ(m)v r (n) (mn) it e(βmn) e mna q .
(1.7)

THE GENERAL SETTING OF THE PROOF

We consider the Dirichlet series

V r (s) = n≥2 c r (n) n s d|n λ (1) d = n≥2 v r (n) n s . (2.1)
This series has the good property to (almost) factor. Note that the summation can be restricted to integers n > z.

Theorem 8. We have

1 + V r (s) = ζ(s)M r s, λ (1) d (2.2) where M r s, λ (1) d = u|r, d≤z 2 uμ(r/u)λ (1) d [u, d] s = 1≤n≤rz 2 h r (n) n s (2.3) with h r (n) = u|r, d≤z 2 [u,d]=n uμ r u λ (1)
d .

(2.4)

One can find a similar decomposition in [18, Lemma 4] with f (n) = 1. Under this condition, we have Ψ r (n) = c r (n) and g(r) = μ 2 (r)/ϕ(r); they are defined between equations ( 9) and [START_REF] Green | The Möbius function is strongly orthogonal to nilsequences[END_REF] there. The factor μ 2 (r) is absent in Motohashi's definition, but the function g will be used only with square-free argument. In this manner, the large sieve inequality given by [START_REF] Motohashi | Primes in arithmetic progressions[END_REF]Lemma 2] reduces to [2, théorème 7A], which is attributed to A. Selberg by E. Bombieri.

Here is the formal identity that gives us a decomposition of 1:

1 = -V r + (1 + V r ).
There follows a decomposition of -ζ /ζ, which we modify with the help of (2.2), and we reach

- ζ ζ = ζ ζ V r -ζ M r . ( 2.5)
This translates into the following pointwise identity:

Λ = -Λ v r + log h r . (2.6)
The corresponding identity for the Möbius function is even more striking:

μ = -μ v r + h r . (2.7)
We finally quote the one for the Liouville function:

λ = -λ v r + ½ X 2 h r .
(2.8) Identities (2.6), (2.7) and (2.8) are the core of our approach. They however still need to be slightly refined, as the variable carried by Λ, μ and λ in the factors Λ v r , μ v r and λ v r , respectively, can be small. This is readily taken care of by a simple truncation (see (2.10)). Let us finally mention that we will average over this family of decompositions.

Let us initiate the proof to fix the notation. The coprimality of our variables with the modulus q will come into play, and we start this section by this point. In situation (V3), the question does not arise. In situation (V4), we write

∼X, ( ,q) =1 Λ( ) it e(β ) e a q ≤ p|q log(2X) log p log p = ω(q) log(2X).
This leads to an error term that is easily absorbed, even numerically on using ω(q) ≤ (log q)/log 2, by all our subsequent error terms. Section 11 explains how to dispense with the coprimality condition in situations (V1) and (V2).

Let us select a square-free integer r ≤ R. We assume that

z 2 R ≤ X.
(2.9)

We further select a (large) parameter M 0 and write (recall (1.6)) S a q , t, β = L (1) r (a, t, β) -L (2) r (a, t, β) -mn∼X, (mn,q)=1 m>M 0 e(βmn)

Λ(m)v r (n) (mn) it e mna q , ( 2.10)
where the first linear form is defined by

L (1) r (a, t, β) = mn∼X, (mn,q)=1 e(βnm) h r (m) log n (nm) it e nma q (2.11)
while the second one is defined by

L (2) r (a, t, β) = mn∼X, (mn,q)=1 m≤M 0 e(βmn) Λ(m)v r (n) (nm) it e mna q .
(2.12)

We now examine the last sum and localize the variables m and n. Notice that n > z. So we start at N = z, go until 2z, etc., until

2 k z ≤ 2X/M 0 < 2 k+1 z, i.e., 0 ≤ k ≤ log(X/(M 0 z))/log 2.
Concerning M , we have N < n ≤ N ≤ 2N , and thus X/(2N ) ≤ X/n < m ≤ 2X/N . So for each N we have two values of M , namely, M 1 = X/(2N ) and M 2 = X/N . We then use the following inequalities, where A(m, n) = e(βmn)Λ(m)v r (n)(nm) -it e(mna/q)½ (mn,q)=1 :

mn∼X A(m, n) 2 = M,N mn∼X, m∼M, n∼N A(m, n) 2 ≤ M,N 1 M,N mn∼X, m∼M, n∼N A(m, n) 2 ≤ 2 log(2X/(M 0 z)) log 2 M,N mn∼X, m∼M, n∼N A(m, n) 2 .
(2.13)

Let us point out that this last summation over m, n has been denoted by S r (a/q, t, β, M, N ) in (1.7). We relax the condition mn ∼ X and remove the coefficient e(βmn) in this sum by appealing to Lemma 45 (see the Appendix) with b = 2, R = X and some δ = δ(M, N ) ∈ (0, 1/2). We find that

S r a q , t, β, M, N = Δ -Δ (mn,q)=1 m∼M, n∼N Λ(m) m i(v+t) v r (n) n i(v+t) e mna q X iv H (v) dv + O * E 1 (δ, r) + E 2 (δ, r) + 2δE 3 (r) (2.14)
where H (v) is provided by Lemma 45 and

E 1 (δ, r) = X<mn≤2 δ X (mn,q)=1 m∼M, n∼N Λ(m)|v r (n)|, E 2 (δ, r) = 2X/2 δ <mn≤2X (mn,q)=1 m∼M, n∼N Λ(m)|v r (n)| (2.15)
and

E 3 (r) = (mn,q)=1 m∼M, n∼N Λ(m)|v r (n)|. (2.16)
The parameter Δ is defined in (A.6) (with b = 2). The proof continues in Section 7.

PREPARATIONS

3.1. A hybrid large sieve inequality. This material is our main ingredient to control the bilinear form arising from our decomposition of the Λ-function.

We quote the following theorem of Selberg from [2, théorème 7A].

Theorem 9. Let N 0 be a given real number. Let (u n ) N 0 <n≤N 0 +N be a sequence of complex numbers. We have

fr≤R, (f,r)=1 f φ(fr) χ mod * f N 0 <n≤N 0 +N u n χ(n)c r (n) 2 ≤ N 0 <n≤N 0 +N |u n | 2 (N + R 2 ),
where c r (m) is the Ramanujan sum modulo r.

The summation is over both coprime variables f and r, subject to fr ≤ R. The parameter N 0 is required since the left-hand side is not (a priori) invariant under translation. We now prove the following hybrid version. Such a result can also be found in [2, théorème 10] and has its origin in [START_REF] Gallagher | A large sieve density estimate near σ = 1[END_REF]; our contribution here is a refined constant 7. Looking more closely at [START_REF] Ramaré | An explicit density estimate for Dirichlet L-series[END_REF]Corollary 6.4], we see that it proves the following.

Theorem 10. Let (u n ) n be a sequence of complex numbers such that n (|u n | + n|u n | 2 ) < ∞. We have d≤D a mod * d T -T n u n n it e na d 2 dt ≤ 7 n |u n | 2 n + D 2 max(T, 10) .
Theorem 11. Let q be some fixed modulus and N 0 be some real number. Let (u n ) n be a sequence of complex numbers such that

n (|u n | + n|u n | 2 ) < ∞.
We have, for any T ≥ 0, r≤R/q, (q,r)=1

1 ϕ(r) a mod q T -T n u n c r (n + N 0 )n it e na q 2 dt ≤ 7 n |u n | 2 n + R 2 max(T, 10) .
Proof. We use

c r (n + N 0 ) = a mod * r e (n + N 0 )a r .
Applying Cauchy's inequality brings us to a position where we can use Theorem 10, since the set {(a/q) + (a /r)} is a subset of the set of points {b/d} with d ≤ R.

Corollary 12. Let q be some fixed modulus. Let (u n ) n be a sequence of complex numbers such that

n (|u n | + n|u n | 2 ) < ∞.
We have, for any T ≥ 0, r≤R/q, (q,r)=1

1 ϕ(r) a mod q T -T n u n c r (n)n it e na q 2 dt 1 + |t| n |u n | 2 n + R 2 log(T + 2) .
Proof. Simply use integration by parts and Theorem 11.

Prime number estimates.

We recall some classical results taken from [START_REF] Rosser | Approximate formulas for some functions of prime numbers[END_REF].

Lemma 13. When M ≥ 101, we have m∼M Λ(m) ≤ 5 4 M.
Proof. Indeed, [START_REF] Rosser | Approximate formulas for some functions of prime numbers[END_REF]Theorem 12] gives us

ψ(x) ≤ 1.04x, x ≥ 0, (3.1)
while [START_REF] Rosser | Approximate formulas for some functions of prime numbers[END_REF]Theorem 10] gives us

ϑ(x) = p≤x log p ≥ 0.84x, x ≥ 101. (3.2)
The lemma follows readily.

We also infer, from [23, corollary to Theorem 1.1],

m≤M 0 Λ(m) m ≤ log M 0 , M 0 ≥ 1. (3.3)
We next need the following extension of the celebrated version of the Brun-Titchmarsh inequality due to H. L. Montgomery and R. C. Vaughan in [START_REF] Montgomery | The large sieve[END_REF]Theorem 2].

Lemma 14. Let d be some integer modulus and a be a reduced residue class modulo d. When

A ≥ B > d ≥ 1, we have A<m≤A+B m≡a[d] Λ(m) log m ≤ 2B ϕ(d) log(B/d) .
We will require the cases d = 1 and d = q. The hypothesis A ≥ B is absent in [START_REF] Montgomery | The large sieve[END_REF]Theorem 2], where only primes are counted. We however need this hypothesis here.

Proof. The hypothesis A ≥ B ensures that if a prime power p k belongs to the interval (A, A + B],

then no other powers of this prime belong to this interval. We may thus bound above our quantity by the number of integers within (A, A + B] that have no prime factors below some parameter z, to which we add the number of primes below z. This upper bound is the one used at the beginning of the proof of [START_REF] Montgomery | The large sieve[END_REF]Theorem 2] in equation (3.3) therein (when B is large) and in Lemma 10 (when B is small). In each occurrence, the summand π(z) accounts for the additional primes (or prime powers) to be included. The proof of [16, Theorem 2] thus applies.

Lemma 15. For any modulus q ≥ 1 and any real number M ≥ max(121, q 3 ), we have

m∼M, m≡a[q] Λ(m) ≤ 9 2 M ϕ(q) .
Proof. By Lemma 14, we find that

m∼M, m≡a[q] Λ(m) ≤ 2 M log(2M ) ϕ(q) log(M/q) .
A numerical application ends the proof.

Lemma 16. For any modulus q ≥ 1 and any real number M ≥ max(121, q 3 ), we have

b mod * q m≡b[q], m∼M Λ(m) 2 ≤ 45 8 M 2 ϕ(q) .
Proof. We collect Lemmas 15 and 13 getting that the left-hand side above is not more than

9 2 M ϕ(q) 5 4 M ≤ 45 8 M 2 ϕ(q) .
Lemma 17. For any modulus q ≥ 1, any ε > 0 and any real number M ≥ q 1+ε , we have

b mod * q m≡b[q], m∼M Λ(m) 2 M 2 ϕ(q) .
Proof. Lemma 14 implies that

m∼M, m≡b[q] Λ(m) ε M ϕ(q)
when M ≥ q 1+ε , and the proof is completed by noticing that

b mod * q m≡b[q], m∼M Λ(m) M.

Möbius function estimates.

The following lemma is quoted from [START_REF] Ramaré | Some elementary explicit bounds for two mollifications of the Moebius function[END_REF].

Lemma 18. When r ≥ 1 and 1.38 ≥ ε ≥ 0, we have

-(1.44 + 5ε + 3.6ε 2 ) ≤ d≤x, (d,r)=1 μ(d) d 1+ε log x d ≤ 1.4 + 4.7ε + 3.3ε 2 + (1 + ε) r 1+ε ϕ 1+ε (r) x ε , where r 1+ε ϕ 1+ε (r) = p|r p 1+ε p 1+ε -1 . ( 3.4)
In the case ε = 0, we have access to better and simpler estimates, and we quote from [31, Corollaries 1.10 and 1.11].

Lemma 19. For any real number x ≥ 1 and any positive integer r, we have

0 ≤ n≤x, (n,r)=1 μ(n) log(x/n) n ≤ 1.00303 r ϕ(r) and 0 ≤ n≤x, (n,r)=1 μ(n) log 2 (x/n) n ≤ 2 log x • r ϕ(r) .
3.4. Explicit averages of some non-negative multiplicative functions. Let us start by recalling estimates on the G-functions. We recall the classical definition

G q (D) = d≤D, (d,q)=1 μ 2 (d) ϕ(d) , G(D) = G 1 (D). (3.5)
We quote from [START_REF] Van Lint | On primes in arithmetic progressions[END_REF]: for any coprime positive integers r and s,

G r (D) ≤ s ϕ(s) G rs (D) ≤ G r (sD), (3.6)
and in particular, when r = 1 and

s = q, G(D) ≤ q ϕ(q) G q (D) ≤ G(qD). (3.7)
According to [START_REF] Ramaré | On Šnirel'man's constant[END_REF]Lemma 3.5] (see also [START_REF] Riesel | On sums of primes[END_REF]),

G(D) ≤ log D + 1.4709, D ≥ 1, (3.8)
and, concerning a lower bound,

log D + 1.06 ≤ G(D), D ≥ 6. (3.9)
We next turn our attention to some of the less studied functions. We appeal to [26, Lemma 3.2], which will be partially recalled during the proof.

Lemma 20. When X ≥ 6, we have

n≤X, (n,6)=1 μ 2 (n) n p|n 1 - 4 p + 8 3p 2 ≤ 0.225(log X + 3.

1).

A bound valid for X ≥ 15 would be enough.

Proof. We define the multiplicative function g by

g(2) = -1 2 , g(3) = -1 3 , g(2 k ) = g(3 k ) = 0 ∀k ≥ 2, g(p) = - 4 p 2 + 8 3p 3 , g(p 2 ) = -1 - 4 p + 8 3p 2 1 p 2 , g(p k ) = 0 ∀k ≥ 3,
so that we have (simply compare the Dirichlet series)

½ (n,6)=1 μ 2 (n) n p|n 1 - 4 p + 8 3p 2 = m=n g( ) 1 m .
Hence the sum S that we need satisfies

S = n≤X m=n g( ) 1 m = ≥1 g( ) m≤X/ 1 m .
We recall the first half of [26, Lemma 3.3], namely,

m≤t 1 m = log t + γ + O * 0.9105 t -1/3
valid for any t > 0. This explains why we can dispense with the condition ≤ X above. We thus get

S = ≥1 g( ) log X + γ + O * 0.9105 X -1/3 ≥1 |g( )| 1/3 .
We readily check that

≥1 |g( )| 1/3 = 1 + 1 2 2/3 1 + 1 3 2/3 p≥5 1 + 4 p 5/3 - 8 3p 8/3 + 1 p 4/3 - 4 p 7/3 + 8 p 10/3 ≤ 11.

Furthermore, with

G(s) = p≥2 1 + k≥1 g(p k ) p ks ,
we can rewrite the above in the form

S = G(0) log X + γ + G (0) G(0) + O * 10 X 1/3 .
We readily find that

G(0) = 1 3 p≥5 1 - 4 p 2 + 8 3p 3 - 1 p 2 + 4 p 3 - 8 3p 4 ≤ 0.225.
On the other hand,

G (0) G(0) = log 2 + log 3 2 - p≥5 (g(p) + 2g(p 2 )) log p 1 + g(p) + g(p 2 ) ≤ 2.42.
Finally, S ≤ 0.225(log X + 3 + 45/X 1/3 ). We used a GP-script to show that

S ≤ 0.225(log X + 3) ∀X ∈ [8, 10 9 ]. (3.10)
As a conclusion,

S ≤ 0.225(log X + 3.1) ∀X ≥ 8, (3.11)
which can be extended to X ≥ 6 by direct inspection.

Lemma 21. When X ≥ 15, we have

≤X, ( ,2)=1 μ 2 ( ) p| 4p 2 -12p + 8 p 2 ≤ 0.0114 X(log X + 3.1) 3 .
A bound valid for X ≥ 15 would be enough. This lemma is not optimal, and more work would yield better constants.

Proof. Let us denote by S the sum to be evaluated. We define the multiplicative functions f 1 and f 2 on primes p ≥ 5 by

f 1 (p) = 4 -12p -1 + 8p -2 = f 2 (p) + 1 and f 1 (p k ) = f 2 (p k ) = 0 as soon as k ≥ 2. At p = 3,
this would give a negative value for f 2 (3), so we majorize f 1 (3) by 1 to keep the non-negativity of f 2 . We thus set f 2 (3 k ) = 0 for every k ≥ 1. We have

f 1 (n) ≤ (f 2 ½)(n). Furthermore, we check that f 2 (m) ≤ (f 3 f 3 f 3 )(m) where f 3 (m) = μ 2 (m)½ (m,6)=1 p|m 1 - 4 p + 8 3p 2 .
(3.12)

More precisely, we have the equality f 2 (m) = (f 3 f 3 f 3 )(m) when m is square-free, while, when m is not square-free, we have f 2 (m) = 0 and (f 3

f 3 f 3 )(m) ≥ 0. Therefore, S ≤ m≤X, (m,2)=1 f 2 (m) ≤X/m 1 ≤ X m≤X, (m,3)=1 f 3 (m) m 3 ≤ 0.0114 X(log X + 3.1) 3 ,
where we have used Lemma 20 for the last inequality.

We define ϕ + (r) = p|r (1 + p).

Lemma 22. We have

r≤X μ 2 (r)ϕ + (r) ϕ(r) ≤ 3.28 X.
This lemma is not optimal and more work would yield better constants.

Proof. The proof is straightforward:

r≤X μ 2 (r)ϕ + (r) ϕ(r) ≤ ≤X μ 2 ( ) p| 2 p -1 |r≤X 1 ≤ X p≥2 1 + 2 p(p -1)
≤ 3.28 X.

3.5. Estimates on the Barban-Vehov weights. Let us start with a rough preliminary estimate.

Lemma 23. We have

d≤z 2 λ (1) d ≤ z 2 /log z.
Proof. We first note that

λ (1) d log z = μ 2 (d) log + z 2 d -log + z d , ( 3.13)
where log + x = max(0, log x). We verify this identity by checking it holds for d ≤ z, for d ∈ [z, z 2 ] and for larger d. We next note that d≤y log(y/d) ≤ y, and the lemma follows readily.

Next we recall [32, Lemma 5.4]; namely, when s > 1 is real, we have

ζ(s) ≤ e γ(s-1) s -1 . (3.14)
Lemma 24. Let R and Q be two positive coprime integers. When ε ∈ [0, 0.168], we have

R|d≤z 2 , (d,Q)=1 λ (1) d d 1+ε ≤ 5(1 + ε) log z Q 1+ε z 2ε R ε ϕ 1+ε (QR)
.

We also have

R|d≤z 2 , (d,Q)=1 λ (1) d d ≤ 1.004 log z Q ϕ(RQ) .
Proof. We first note that (cf. (3.13))

λ ( 1
)
d log z = μ(d) log + z 2 d -μ(d) log + z d . (3.15)
We prove this identity again by checking it holds for d ≤ z, for d ∈ [z, z 2 ] and for larger d. On using the decomposition (3.15), we find that

R|d≤z 2 , (d,Q)=1 λ (1) d log z d 1+ε = R|d≤z 2 , (d,Q)=1 μ(d) log(z 2 /d) d 1+ε - R|d≤z, (d,Q)=1 μ(d) log(z/d) d 1+ε = μ(R) R 1+ε ≤z 2 /R ( ,QR)=1 μ( ) log((z 2 /R)/ ) 1+ε - μ(R) R 1+ε ≤z/R ( ,QR)=1 μ( ) log((z/R)/ ) 1+ε .
We use Lemma 18 to show that the absolute value of the left-hand side is not more than

μ 2 (R) R 1+ε 1.4 + 4.7ε + 3.3ε 2 + (1 + ε) (RQ) 1+ε ϕ 1+ε (RQ) z 2ε R ε + 1.44 + 5ε + 3.6ε 2 .
Since (2.84 + 9.7ε + 6.9ε 2 )/(1 + ε) ≤ 4 when ε ∈ [0, 0.168], the first inequality of the lemma follows. The second inequality follows by using the same decomposition (3.15) and invoking Lemma 19.

Lemma 25. Let δ be a given integer and ε ∈ (0, 0.16]. We have

d 1 ,d 2 ≤z 2 , δ|[d 1 ,d 2 ] λ (1)
d 1 λ (1) d 2 [d 1 , d 2 ] 1+ε ≤ 25(1 + ε) 2 z 4ε ζ(1 + ε) δ ε log 2 z p|δ 3 p -1 .
Proof. Let us denote by S the sum to be studied. We find that

S = δ 1 δ 2 δ 3 =δ δ 1 δ 3 |d 1 ≤z 2 δ 2 δ 3 |d 2 ≤z 2 (d 1 ,δ 2 )=(d 2 ,δ 1 )=1 λ (1) d 1 λ (1) d 2 [d 1 , d 2 ] 1+ = δ 1 δ 2 δ 3 =δ δ 1+ 3 ≤z 2 ( ,δ)=1 ϕ 1+ ( ) δ 1 δ 3 |d 1 ≤z 2 δ 2 δ 3 |d 2 ≤z 2 (d 1 ,δ 2 )=(d 2 ,δ 1 )=1 λ (1) d 1 λ (1) d 2 (d 1 d 2 ) 1+
obtained by using

1 [d 1 , d 2 ] 1+ = (d 1 , d 2 ) 1+ (d 1 d 2 ) 1+ = δ 1+ε 3 (d 1 d 2 ) 1+ |d 1 /δ 3 , |d 2 /δ 3 ϕ 1+ ( )
and by noticing that is prime to δ; this implies that, in fact, | d 1 /(δ 1 δ 3 ) and | d 2 /(δ 2 δ 3 ). We apply Lemma 24 twice (once to the sum over d 1 and once with the sum over d 2 ) to get

|S| ≤ 25(1 + ) 2 z 4 δ 1+ (log z) 2 δ ϕ 2 1+ (δ) δ 1 δ 2 δ 3 =δ 1 δ 3 ≤z 2 , ( ,δ)=1 μ 2 ( ) 2 ϕ 1+ ( ) .
But we have

≥1, ( ,δ)=1 μ 2 ( ) ϕ 1+ ( ) = (p,δ)=1 1 + 1 p 1+ -1 = ϕ 1+ (δ) δ 1+ ζ(1 + ).
With this we get

|S| ≤ 25(1 + ) 2 z 4 ζ(1 + ) (log z) 2 δ ϕ 1+ (δ) δ 1 δ 2 δ 3 =δ 1 δ 3 .
Finally we can see that

1 ϕ 1+ (δ) δ 1 δ 2 δ 3 =δ 1 δ 3 = 1 ϕ 1+ (δ) δ 3 |δ 2 ω(δ/δ 3 ) δ 3 = 1 ϕ 1+ (δ) p|δ 2 1 + 1 2p ≤ p|δ 3 p -1 ,
from which the proof follows.

Lemma 26. Let δ be a given integer and ε ∈ (0, 0.16]. We have

d 1 ,d 2 ≤z 2 λ ( 1
)
d 1 λ (1) d 2 [δ, d 1 , d 2 ] 1+ε ≤ 25(1 + ε) 2 z 4ε ζ(1 + ε) log 2 z p|δ 4 p .
Proof. We use Selberg's diagonalization process as usual and appeal to Lemma 25 to get

d 1 ,d 2 ≤z 2 λ ( 1
)
d 1 λ (1) d 2 [δ, d 1 , d 2 ] 1+ ≤ t|δ ϕ 1+ (t) δ 1+ d 1 ,d 2 ≤z 2 t|[d 1 ,d 2 ]
λ

(1)

d 1 λ (1) d 2 [d 1 , d 2 ] 1+ ≤ 25(1 + ) 2 z 4 ζ(1 + ) δ 1+ log 2 z t|δ ϕ 1+ (t) t p|t 3 p -1 . But we have 1 δ 1+ t|δ ϕ 1+ (t) t p|t 3 p -1 ≤ 1 δ t|δ ϕ 1+ (t) t 2 p|t 3 p -1 ≤ p|δ 4 p ,
where we have used the following to get the second inequality:

ϕ 1+ (t) t 2 ≤ ϕ(t).
The next lemma is contained in the main theorem of [START_REF] Betah | Explicit expression of a Barban & Vehov theorem[END_REF].

Lemma 27. When B ≥ z ≥ 100, we have

n≤B d|n λ (1) d 2 n ≤ 166 log B log z .
In the next two lemmas, we deal with

A r = u|r, d≤z 2 (ud,q)=1 uμ(r/u)λ (1) d [u, d] and A r = u|r, d≤z 2 (ud,q)=1 uμ(r/u) log([u, d])λ (1) d [u, d] . ( 3.16)
We first get simpler expressions. The parameter r will always be square-free and prime to q, but we prefer to repeat it when necessary.

Lemma 28. When r is square-free and prime to q, we have

A r ϕ(r) = r|d≤z 2 , (d,q)=1 λ (1) d d as well as A r ϕ(r) = r|d≤z 2 , (d,q)=1 log(rd) λ (1) d d + |r Λ( ) -1 (r/ )/d≤z 2 , (d,q)=1 λ (1) d d - r|d≤z 2 , (d,q)=1 λ (1) d d .
Proof. For this proof, it will save on typographical work to define f d to be λ

(1)
d /d when d is prime to q and 0 otherwise. The case of A r is easily dealt with. We simply write

A r = u|r, d≤z 2 uμ(r/u)df d [u, d] = δ|r ϕ(δ) δ|u|r, δ|d≤z 2 μ r u f d = ϕ(r) r|d≤z 2 f d ,
and this ends the proof. Concerning A r , we use [u, d] = ud/(u, d) to get

A r = u|r, d≤z 2 (ud,q)=1 uμ(r/u) log([u, d])df d [u, d] = u|r, d≤z 2 (ud,q)=1 μ r u (u, d)f d log(ud) -log((u, d)) = B -C
say. This calls for the study of two partial quantities, B and C:

B = δ|u|r, δ|d≤z 2 ϕ(δ)μ r u f d log d + log u δ + log δ = ϕ(r) r|d≤z 2 log d • f d + δ|r, δ|d≤z 2 ϕ(δ)Λ r δ f d + ϕ(r) log r r|d≤z 2 f d ,
since μ log = Λ. Moreover, r is square-free, and this implies ϕ(δ) = ϕ(r)/ϕ(r/δ). On using = r/δ, we have

B = ϕ(r) r|d≤z 2 log d • f d + ϕ(r) |r, (r/ )|d≤z 2 Λ( ) -1 f d + ϕ(r) log r r|d≤z 2 f d . The partial quantity C is (use log((u, d)) = |u, |d Λ( )) C = u|r, d≤z 2 μ r u log((u, d))(u, d)f d = |u|r, |d≤z 2 Λ( )μ r u (u, d)f d = ϕ(r) |r Λ( ) -1 r|d≤z 2 f d .
The last equality asks for some details: being fixed dividing r and d, we have

|u|r μ r u (u, d) = v|r μ r v (v, d ),
where r = r/ and d = d/ . This last sum vanishes if r d and has value ϕ(r ) = ϕ(r)/ϕ( ) otherwise. Hence

A r ϕ(r) = B ϕ(r) - C ϕ(r) = r|d≤z 2 , (d,q)=1 log(rd) f d - |r Λ( ) -1 (r/ )|d≤z 2 f d - r|d≤z 2 f d .
The lemma follows readily. Lemma 29. When r is square-free and prime to q, we have

r|d≤z 2 , (d,q)=1 log 4X erd λ (1) d d ≤ 1.004 log(4X/(er)) log z + 2 q ϕ(rq) .
Proof. We appeal to the decomposition given by (3.15) and find that

r|d≤z 2 , (d,q)=1 log 4X erd λ (1) d d = r|d≤z 2 , (d,q)=1 log 4X erd μ(d) log(z 2 /d) d log z - r|d≤z, (d,q)=1 log 4X erd μ(d) log(z/d) d log z = r|d≤z 2 , (d,q)=1 log 4X erz 2 μ(d) log(z 2 /d) d log z + r|d≤z 2 , (d,q)=1 μ(d) log 2 (z 2 /d) d log z - r|d≤z, (d,q)=1 log 4X erz μ(d) log(z/d) d log z - r|d≤z, (d,q)=1 μ(d) log 2 (z/d) d log z .
Lemma 19 tells us that each sum is signed and is bounded. For instance, when μ(r) = 1, the first two terms are non-negative while the next two are non-positive, and similarly when μ(r) = -1. It is thus enough to bound each term. We further note, with α = 1.003003, that

α log 4X erz 2 + 2 log z 2 r ≤ α log 4X erz 2 + 4 log z = α log 4X er + (4 -2α) log z.
The lemma follows readily.

3.6. Handling some smooth sums. The study of the linear parts relies on the exact evaluation of smooth sums: we gather this material here.

Lemma 30. When M and N > 0 are real numbers such that M + N ≥ 1 and a is an integer prime to q, we have

M<n≤M +N (n,q)=1 e na q = μ(q)N q + O * (ϕ(q)).
Proof. We split the interval (M, M + N ] into N/q + O * (1) intervals containing q consecutive integers and a final interval containing, say, h integers prime to q. Since h ≤ ϕ(q) -1, the lemma is proved.

By integration by parts, we get n≤N, (n,q)=1 e na q log n = n≤N, (n,q)=1 e na q log N -N 1 n≤t, (n,q)=1 e na q dt t .

We use this formula for 2N and N and get

N<n≤2N (n,q)=1 e na q log n = n≤N (n,q)=1 e na q log 2 + N<n≤2N (n,q)=1 e na q log(2N ) - 2N N n≤t (n,q)=1 e na q dt t .
Hence we get the following lemma. Lemma 31. When N is a real number and a is an integer prime to q, we have N<n≤2N, (n,q)=1 e na q log n = μ(q)N log(4N/e) q + O * ϕ(q) log(8N ) .

Lemma 32. When M and N > 0 are real numbers and b is an integer, we have

M<n≤M +N n≡b[q]
e(βn)

n it = 1 q M +N M e(βv) dv v it + O (|t| + |β|(M + N ) + 1) log(2(M + N ))
and

M<n≤M +N n≡b[q] e(βn) log n n it = 1 q M +N M e(βv) log v v it dv + O (|t| + |β|(M + N ) + 1) log 2 (2(M + N )) .
Proof. We define f 1 (α, ) = e(βq )/(α + ) it and f 2 (α, ) = log(α + )f 1 (α, ) for α = b/q with 1 ≤ b ≤ q -1 and first study, for

f = f 1 or f = f 2 , S(L; f ) = 1≤ ≤L f ( ).
(3.17)

We have

S(L; f ) = - L 1 [u]f (u) du + [L]f (L) = f (1) + L 1 f (u) du + O ⎛ ⎝ L 1 |f (u)| du + |f (L)| ⎞ ⎠ = L 1 f (u) du + O (|t| + βqL) log 2 (2L) .
We consider

S M + N -b q ; f 1 -S M -b q ; f 1 log q q it + S M + N -b q ; f 2 -S M -b q ; f 2 1 q it
and, as a consequence, we find that

MT ⎛ ⎜ ⎝ M<n≤M +N n≡b[q] e(βn) log n n it ⎞ ⎟ ⎠ = (M +N -b)/q (M -b)/q e(β(b + uq)) log(b + uq) (b + uq) it du = 1 q M +N M e(βv) log v v it dv,
where MT(•) means the main term of the parenthesized quantity. The lemma follows readily.

Lemma 33. When M and N > 0 are real numbers and a is an integer prime to q, we have

M<n≤M +N (n,q)=1 e(βn) e(an/q) n it = μ(q) q M +N M e(βv) dv v it + O q(|t| + |β|(M + N ) + 1) log(2(M + N ))
and

M<n≤M +N (n,q)=1 e(βn) log n n it e an q = μ(q) q M +N M e(βv) log v v it dv + O q(|t| + |β|(M + N ) + 1) log 2 (2(M + N )) .
Proof. This is a simple exercise with the previous lemma.

THE FIRST LINEAR FORM

We study here the first linear form defined in (2.11), and this section is devoted to proving Lemma 34 and 35.

The case

t = β = 0.
Lemma 34. When Rz 2 /q ≤ X and (a, q) = 1, we have

r≤R/q, (r,q)=1 μ 2 (r) L (1) r (a, 0, 0) ϕ(r) ≤ X q G(R) 3.012 log(4X/e) log z + 2 + 3.3ϕ(q) R q z 2 log z log(8X).
Proof. We start from (2.11) and sum over n first by using Lemma 31; we find that

L (1) r (a, 0, 0) = μ(q) q X m≤2X (m,q)=1 h r (m) log(4X/(me)) m + O * ⎛ ⎜ ⎝ ϕ(q) log(8X) m≤2X (m,q)=1 |h r (m)| ⎞ ⎟ ⎠ . (4.1)
The bound m ≤ 2X can be replaced in both summations by m ≤ rz 2 since h r (m) vanishes otherwise. Note also that rz 2 ≤ 2X. We set

A r = m≤rz 2 , (m,q)=1 h r (m) log(4X/(me)) m = A r log 4X e -A r , ( 4.2)
on recalling (3.16).

Bounding the main term in (4.1). As a consequence, we deduce the following. We combine (3.16) with Lemma 28 to infer that

A r ϕ(r) = log 4X e r|d≤z 2 (d,q)=1 λ (1) d d - r|d≤z 2 (d,q)=1 log(rd) λ (1) d d - |r Λ( ) -1 ⎛ ⎜ ⎝ (r/ )|d≤z 2 (d,q)=1 λ (1) d d - r|d≤z 2 (d,q)=1 λ (1) d d ⎞ ⎟ ⎠ = r|d≤z 2 (d,q)=1 log 4X erd λ (1) d d - |r Λ( ) -1 ⎛ ⎜ ⎝ (r/ )|d≤z 2 (d,q)=1 λ (1) d d - r|d≤z 2 (d,q)=1 λ (1) d d ⎞ ⎟ ⎠ .
In this form, Lemma 29 and the second part of Lemma 24 (with R = r/ and Q = q) apply directly to yield the bound We first notice that r ≤ 4X/e and sum over r. Recalling that (3.7) implies (q/ϕ(q))G q (R/q) ≤ G(R), we finally get

|A r | ϕ(r) ≤ 1.
r≤R/q, (r,q)=1 μ 2 (r)|A r | ϕ(r) ≤ G(R)
r≤R/q, (r,q)=1 μ 2 (r) ϕ(r) m≤2X, (m,q)=1 |h r (m)| ≤ z 2 log z r≤R/q, (r,q)=1 μ 2 (r) δ|r 2 ω(δ) ϕ(δ) ≤ R q z 2 log z p≥2 1 + 2 p(p -1) ≤ 3.3 R q z 2 log z .
The proof of Lemma 34 is complete.

The general case.

Lemma 35. When Rz 2 /q ≤ X, we have

r≤R/q, (r,q)=1 μ 2 (r) L (1) r (a, t, β) ϕ(r) ≤ X q G(R) 2.008 log(4XR/(eq)) log z + 3 + O (|t| + |β|X + 1) log 2 X Rz 2 log z .
Proof. We start from (2.11) and sum over n first by using Lemma 33; we find that

L (1) r (a, t, β) = μ(q) q m≤2X, (m,q)=1 h r (m) m it 2X/m X/m e(βmv) log v v it dv + O q(|t| + |β|X + 1) log 2 X m≤2X, (m,q)=1
|h r (m)| .

PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS Vol. 314 2021

The change of variable w = vm yields

L (1) r (a, t, β) = μ(q) q m≤2X, (m,q)=1 h r (m) m ⎛ ⎝ 2X X e(βw) log w w it dw - 2X X e(βw) dw w it log m ⎞ ⎠ + O q(|t| + |β|X + 1) log 2 X m≤2X, (m,q)=1
|h r (m)| so that, with the notation A r and A r from (3.16),

L (1) r (a, t, β) = μ(q) q 2X X e(βw) log w w it dw A r - μ(q) q 2X X e(βw) dw w it A r + O q(|t| + |β|X + 1) log 2 X m≤2X, (m,q)=1 |h r (m)| .
From then onwards, the treatment of

L (1)
r (a, t, β) can be mimicked with the one of

L (1)
r (a, 0, 0). We leave the details to the reader.

THE SECOND LINEAR FORM

We study here the second linear form defined in (2.12). Lemma 36. When a and r are such that (a, q) = (r, q) = 1, we have L (2) r (a, 0, 0) ≤ 1.004

μ 2 (q)X log M 0 ϕ(q) log z + 1.04ϕ(q)ϕ + (r)M 0 z 2 log z with ϕ + (r) = |r = p|r (1 + p). Proof.
We readily find that L (2) r (a, t, β)

= m≤M 0 (m,q)=1 Λ(m) m it d≤z 2 (d,q)=1 λ (1) d n∼X/m (n,q)=1, d|n e(βmn) c r (n) n it e mna q = m≤M 0 (m,q)=1 Λ(m) m it d≤z 2 (d,q)=1 λ (1) d |r μ r n∼X/m (n,q)=1, [d, ]|n e(βmn)
e(mna/q) n it .

(5.1)

Now we specialize to t = β = 0, apply Lemma 30 and get

L (2) r (a, 0, 0) = m≤M 0 (m,q)=1 Λ(m) d≤z 2 (d,q)=1 λ (1) d |r μ r μ(q)X m[d, ]q + O * (ϕ(q)) = μ(q)Xϕ(r) q m≤M 0 (m,q)=1 Λ(m) m d≤z 2 (d,q)=1, r|d λ (1) d d + O * 1.04ϕ(q)ϕ + (r)M 0 z 2 log z ,
since the error in the above equation is

ϕ(q) m≤M 0 Λ(m) d≤z 2 λ (1) d |r ≤ 1.04ϕ(q)M 0 z 2 log z ϕ + (r),
where the sum over m is treated via (3.1) and the sum over d is treated via Lemma 23. We appeal to the second estimate of Lemma 24 (with R = r and Q = q) to get

L (2) r (a, 0, 0) ≤ 1.004μ 2 (q)X log M 0 ϕ(q) log z + 1.04ϕ + (r)ϕ(q)M 0 z 2 log z ,
where the sum over m is this time treated via (3.3). The lemma follows readily.

Adapting this proof to the general case from (5.1) is not difficult. We get Lemma 37. When a and r are such that (a, q) = (r, q) = 1, we have

L (2) r (a, t, β) μ 2 (q)X log M 0 (1 + |t|)ϕ(q) log z + O q(|t| + |β|X + 1)ϕ + (r)M 0 z 2 log z log X .
We quote the next direct consequence of this lemma.

Lemma 38. For any ε > 0, we have

T -T r≤R/q, (r,q)=1 μ 2 (r) ϕ(r) L (2) r (a, t, 0) dt ε X(log T )(log R) log M 0 ϕ(q) log z + M 0 z 2 T 2 R 1+ε log X log z .

THE ERROR TERM DUE TO THE SEPARATION OF VARIABLES

Lemma 39. When ε ∈ (0, 0.154] and R/q ≥ 15, we have

rq≤R (r,q)=1 μ 2 (r) ϕ(r) n≤B (n,q)=1 |v r (n)| 2 n ≤ 0.285 G q R q (1 + ε) 2 B ε z 4ε ζ 2 (1 + ε)R q log 2 z log R q + 3.1 3 .
Proof. We use

|c r (n)| 2 ≤ ϕ((n, r)) 2 = δ|n, δ|r δf 2 (δ),
where f 2 (δ) = p|δ (p -2). Thus, on using (3.6),

r≤R/q (r,q)=1 μ 2 (r)|c r (n)| 2 ϕ(r) = δ|n δf 2 (δ) δ|r≤R/q (r,q)=1 μ 2 (r) ϕ(r) = δ|n f 2 (δ)G δq R δq ≤ δ|n δ≤R/q f 3 (δ)G q R q ,
where f 3 (δ) = μ 2 (δ) p|δ (p -1)(p -2)/p. On denoting by S the sum to be bounded above, we get

S ≤ rq≤R (r,q)=1 μ 2 (r) ϕ(r) n≤B d|n λ (1) d 2 |c r (n)| 2 n ≤ G q R q δ≤R/q f 3 (δ) δ|n≤B 1 n d|n λ (1) d 2 ≤ G q R q B ε δ≤R/q f 3 (δ) δ|n 1 n 1+ε d|n λ (1) d 2 ≤ G q R q B ε ζ(1 + ε) δ≤R/q f 3 (δ) d 1 ,d 2 ≤z 2 λ ( 1
)
d 1 λ (1) d 2 [δ, d 1 , d 2 ] 1+ε .

PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS Vol. 314 2021

We appeal to Lemma 26 and (3.14):

S G q (R/q) ≤ 25(1 + ε) 2 B ε z 4ε ζ 2 (1 + ε) log 2 z δ≤R/q f 3 (δ) p|δ 4 p .
A use of Lemma 21 gives us

S ≤ G q R q 25(1 + ε) 2 B ε z 4ε ζ 2 (1 + ε)R q log 2 z • 0.0114 log R q + 3.1 3 .
The lemma follows readily.

PROOF OF THE EXPLICIT THEOREM 7

We continue the argument started in Section 2.

7.1. Preparation of each |S r (a/q, t, β, M, N )|. Recall the decomposition (2.14) of S r . We first have the next explicit bound.

Lemma 40. When M ≥ 10 14 and δ ≥ M -1/4 , we have

E 1 (δ, r) + E 2 (δ, r) + 2δE 3 (r) ≤ 10.5 δM n∼N, (n,q)=1 |v r (n)|.
When ignoring the explicit aspect, here is what we get.

Lemma 41. Let > 0. When δM M , we have

E 1 (δ, r) + E 2 (δ, r) + 2δE 3 (r) δM n∼N, (n,q)=1 |v r (n)|.
Proof. We first notice that, when N < n ≤ 2N , M < m ≤ 2M and X < mn ≤ 2 δ X, we check that, with

A = max(M, X/n), max M, X n , min 2M, 2 δ X n ⊂ A, A + (2 δ -1)M ⊂ A, A + (2δ log 2)M , since 2 δ -1 = δ log 2 0 e u du ≤ 2δ log 2. Similarly, concerning the conditions N < n ≤ 2N , M < m ≤ 2M and 2 1-δ X < mn ≤ 2X, we check that, with A = max(M, 2 1-δ X/n), max M, 2 1-δ X n , min 2M, 2X n ⊂ A, A + 2(1 -2 -δ )M ⊂ A, A + (2δ log 2)M , since this time 1 -2 -δ = 0 -δ log 2 e u du ≤ δ log 2.
We note that 2 log 2 ≤ 1.4. By Lemma 14 applied twice, we find that

E 1 (δ, r) + E 2 (δ, r) ≤ 2 2 • 1.4δM log(1.4δM ) log(2M ) n∼N |v r (n)|.
Note that, since δ ≥ M -1/4 , we have δM ≥ M 3/4 and

2 • 1.4 log(2M ) log(1.4M 3/4 ) ≤ 3.8,
and thus

E 1 (δ, r) + E 2 (δ, r) ≤ 3.8 δM n∼N |v r (n)|. (7.1)
Concerning E 3 (r), we use Lemma 13, getting

E 3 (r) ≤ 5 4 M n∼N |v r (n)|. (7.2)
Note that 2 • 3.8 + 2 • 5/4 = 10.1 ≤ 10.5. Hence, on using the classical inequality |a 2 . We use Lemma 16 provided that M ≥ max(121, q 3 ):

+ b| 2 ≤ 2(|a| 2 + |b| 2 ), S r a q , t, β, M, N 2 ≤ 2 Δ -Δ b mod * q m≡b[q], m∼M Λ(m) m i(v+t) 2 |H (v)| 2 dv × Δ -Δ b mod * q (n,q)=1, n∼N v r (n) n i(v+t) e nba q 2 dv + 2 10.5 δM n∼N |v r (n)| 2 . (7.3) By Lemma 45 (see the Appendix), we have R | H(δ, λ, κ; v)| 2 dv = R |H(δ, λ, κ; u)| 2 du = (2 -2δ) × (log 2) 2 /(4π)
S r a q , t, β, M, N 2 ≤ 2 • 2 (log 2) 2 (4π) 2 45 8 M 2 ϕ(q) Δ -Δ b mod * q (n,q)=1, n∼N v r (n) n i(v+t) e nb q 2 dv + 221δ 2 M 2 (N + 1) n∼N |v r (n)| 2 ;
i.e., since MN ≤ X and N ≥ z ≥ 121,

S r a q , t, β, M, N 2 ≤ 45 2 (log 2) 2 (4π) 2 M 2 ϕ(q) Δ -Δ b mod * q (n,q)=1, n∼N v r (n) n i(v+t) e nb q 2 dv + 446δ 2 X 2 n∼N |v r (n)| 2 n . ( 7.4)
7.2. Average estimate (over r and (M, N )) of |S r (a/q, t, β, M, N )| 2 . Let us use the shorter notation

Σ(M, N ) = rq≤R, (r,q)=1 μ 2 (r) ϕ(r) S r a q , t, β, M, N 2 . ( 7.5)
We sum (7.4) over r and use the large sieve inequality from Theorem 11 to get (since MN ≤ X)

Σ(M, N ) ≤ 158 M 2 ϕ(q) (log 2) 2 (4π) 2 (n,q)=1 n∼N d|n λ (1) d 2 (n + R 2 Δ) + 446δ 2 X 2 rq≤R (r,q)=1 μ 2 (r) ϕ(r) n∼N |v r (n)| 2 n .
We take R = z 1/4 = 2δ -1 > 2q 2 and M 0 = z ≥ q 8 , with z ≥ 10 14 . We assume that β = 0, and thus (see (A.6))

δ 2 Δ = 2 π log 2 . Thus n + R 2 Δ ≤ 2N + 2δ -2 π log 2 • 4δ -2 ≤ 2N + 4δ -4 ≤ 2N + z 4 . This is since R = 2δ -1 .
Furthermore, for any n ∈ (N, 2N ], we have n(2N

+ z/4) ≤ 4.5N 2 since N ≥ z. Hence Σ(M, N ) ≤ 4.4 X 2 ϕ(q) (n,q)=1, n∼N d|n λ (1) d 2 n + 446δ 2 X 2 rq≤R, (r,q)=1 μ 2 (r) ϕ(r) n∼N |v r (n)| 2 n .
It is time to sum over relevant (M, N ). We set

Σ = M,N Σ(M, N ). (7.6)
Summation over M is readily dealt with: there are at most two M 's for each N . Thus

Σ ≤ 8.8 X 2 ϕ(q) (n,q)=1, n≤2X/z d|n λ (1) d 2 n + 892δ 2 X 2 rq≤R, (r,q)=1 μ 2 (r) ϕ(r) n≤2X/z |v r (n)| 2 n . ( 7.7)
We forget the condition (n, q) = 1 in the first summation and appeal to Lemma 27, while we use Lemma 39 for the second term. All of that yields

Σ ≤ 8.8 X 2 ϕ(q) • 166 log(2X/z) log z + 892δ 2 X 2 G q R q 0.285(1 + ε) 2 (2e 2γ Xz 3 ) ε R ε 2 q log 2 z log R q + 3.1 3 . Hence (since δ = 2R -1 , ϕ(q) ≤ q, G q (R/q) ≤ G(R/q) and R/q ≥ R 1/3 = X 1/48 ≥ 15) ϕ(q) X 2 Σ ≤ 1461 log(2X/z) log z + 1016 (1 + ε) 2 (2e 2γ Xz 3 ) ε Rε 2 log 2 z log R q + 3.1 4 .
We used equation (3.8). Now we directly choose z = X 1/4 and ε = 8/(7 log X) ≤ 0.009 (since X ≥ 250 24 ). We thus get ϕ(q) X 2 Σ ≤ 4414 + 95 200 (log Rlog q + 3.1) 4 R .

Next, with R = z 1/4 ≥ q 3/2 ≥ 250 3/2 , Σ ≤ 32 830 X 2 ϕ(q) . (7.8)

Conclusion of the proof of Theorem 7.

Recall that z = X 1/4 ≥ q 6 and assume that q ≥ 250. We have

G q R q S a q , t, β ≤ r≤R/q, (r,q)=1 μ 2 (r) ϕ(r) |L (1) r (a, β, t)| + |L (2) r (a, β, t)| + G q R q 32 830 X 2 ϕ(q) 2 log(2X/z 2 ) log 2 1/2
, where the factor 2(log(2X/z 2 ))/log 2 comes from (2.13) since M 0 = z. This bound is valid for β = 0 (because we specialized Δ) and arbitrary t. We continue numerically by specializing further t = 0. We select R 4 = z = M 0 = X 1/4 ≥ q 6 and q ≥ 250.

Concerning L

(1) r , we use Lemma 34: the hypothesis Rz 2 /q ≤ X is met. We thus get r≤R/q, (r,q)=1 μ 2 (r) ϕ(r) L (1) r (a, 0, 0)

≤ X q G(R) 3.012 log(4X/e) log z + 2 + 3.3ϕ(q) R q z 2 log z log(8X) ≤ X q G(R) 3.012 log(4X/e) (1/4) log X + 2 + 3.3X -19/48 (1/16) log X + 1.06 log(8X) (1/4) log X ≤ 16 X q G(R).
We use (3.9) on G(R), which is allowed since R = X 1/16 ≥ 6. Concerning L

(2) r , we use Lemmas 36 and 22, getting (we check that ϕ(q)M 0 Rz 2 ≤ X 107/192 ) r≤R/q, (r,q)=1 μ 2 (r) ϕ(r) L (2) r (a, 0, 0

) ≤ 1.004 G q R q μ 2 (q)X log M 0 ϕ(q) log z + 1.04ϕ(q)M 0 z 2 log z • 3.28 R q ≤ G(R) X q 1.004 + 3.4112 X 107/192-1 ((1/16) log X + 1.06)(1/4) log X ≤ 1.005 G(R) X q .
This finally amounts to

18 G(R) X q + G q R q 47 860 X 2 ϕ(q) log X 1/2 .
We finally get (R/q ≥ R 1/3 = X 1/48 and G q (R/q) ≥ (ϕ(q)/q)G(R/q))

S a q , 0, 0 ≤ 1292 √ q X ϕ(q)
provided 250 ≤ q ≤ X 1/24 .

PROOF OF THEOREM 4 IN VERSION (V4)

Proving Theorem 4 with primes rather than with the Möbius function is a simple modification of the proof of Theorem 7. Let us recall (2.13) from Section 2:

S a q , t, β -L (1) r (a, t, β) + L (2) r (a, t, β) 2 ≤ 2 log(2X/(M 0 z)) log 2 × M,N ⎛ ⎜ ⎝ Δ -Δ (mn,q)=1 m∼M, n∼N Λ(m) m i(v+t) v r (n) n i(v+t) e mna q |H (v)| dv + E 1 (δ, r) + E 2 (δ, r) + 2δE 3 (r) ⎞ ⎟ ⎠ 2 . (8.1)
This is so because mn∼X A(m, n) is nothing other than S(a/q, t, β) -L

(1)

r (a, t, β) + L (2)
r (a, t, β). The localized version of this quantity, i.e., with m ∼ M and n ∼ N , is S r (a/q, t, β, M, N ). However, in this latter sum, the variables m and n are still constrained by the condition mn ∼ X. We then only have to introduce (2.14) in (2.13) to get the above.

The only remaining difficulty is to collect the diverse conditions on the parameters R, δ, M 0 , z, q and X. We list them below (we simplify them: we neglect the difference between ϕ(q) and q, we also require that R 2 /(zδ 2 ) ≤ 1 while an upper bound of any constant would also do, and so on). For some > 0, we need

(1) R ≥ 8q; (2) M 0 , R, z ≥ X ;
(3) δM 0 ≥ M 0 for the separation of variables by Lemma 41; (4) δ 2 R log 6 X ≤ 1 for the large sieve argument on the remainder term coming from the separation of variables: indeed, by Lemmas 39 (with ε = 1/log z 1/log N ) and 41, we find that

r≤R/q μ 2 (r) ϕ(r) E 1 (δ, r) + E 2 (δ, r) + 2δE 3 (r) 2 δ 2 (MN ) 2 (log R) 4 R q .
After summation over N as a power of 2 and over M (at most two values), this bound is multiplied by log X. The factor O(log(X/M 0 z)) in front introduces another logarithm;

PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS Vol. 314 2021

(5) (βX) 2 ≤ δ -1 , and this ensures that Δ, which is given by Δ = 100(δ -1 + (βX) 2 )/δ, satisfies Δ δ -2 ;

(6) R 2 /(zδ 2 ) ≤ 1 and M 0 ≥ q 1+ for the large sieve argument on the bilinear form: indeed, we proceed as in (7.3) and apply Corollary 12. The final L 2 -norm is handled by Lemma 27. We need R 2 Δ z, and that is granted by R 2 δ -2 z since N starts at z. We employ Lemma 17 rather than Lemma 16 for its less stringent condition M ≥ q 1+ε . Since M can be as small as M 0 , this explains the second condition above; [START_REF] Fouvry | Exponential sums with monomials[END_REF]

(|t| + 1 + |β|X)
√ qRz 2 ≤ X for the term coming from L √ qRz 2 M 0 ≤ X for the term coming from L

(2) r (a, t, β) by Lemma 37. Condition ( 7) is a consequence of condition [START_REF] Gallagher | A large sieve density estimate near σ = 1[END_REF]. It is better to choose M 0 as small as possible, so let us select M 0 = min(q, δ -1 )X 2θ for some positive θ. We want the ranges in β and t to be as large as possible, so we want δ and z to be small. However, condition ( 6) is imposed to reach a balance between these two aims. It is also better to choose R as small as we may, so we choose R = qX 2θ and z = q 2 X 4θ /δ 2 .

Let us choose M 0 = qX 4θ and δ = q -1 X -2θ . We also assume that |βX| ≤ √ q X θ and (1 + |t| + |β|X)q 13/2 X 22θ ≤ X.

We set η = (2 -44θ)/13 to reach the statement of the theorem.

PROOF OF THEOREM 2 IN VERSION (V4)

The proof of the L 1 -Theorem 2 is similar to the proof of Theorem 7 with one notch of difficulty added. The required modifications are however rather simple; we explain them in this section. The case of the primes is slightly more difficult as it entails using the Barban-Davenport-Halberstam theorem, while the bounds |λ(m)|, |μ(m)| ≤ 1 will be enough in the other cases. We will rely on the bound |H (v)|

(1 + X|β|)/(1 + |v|) with β = 0, combined with Corollary 12. We start with

G q R q T -T ∼X, ( ,q)=1 Λ( ) it e a q dt
T -T r≤R/q, (r,q)=1 μ 2 (r) ϕ(r) L (1) r (a, t, 0) + L (2) r (a, t, 0) dt

+ T -T r≤R/q, (r,q)=1 μ 2 (r) ϕ(r) mn∼X (mn,q)=1, m>M 0 Λ(m)v r (n) (mn) it e mna q dt.
The term with L

(1) r on the right-hand side is estimated as in Lemma 35. The term with L

(2)

r is estimated as in Lemma 38. The estimation of the third term on the right-hand side requires a modification at the level of equation (7.3). We first split the summation in (m, n) according to their size (M, N ) and then apply Cauchy's inequality on the resulting sums, getting

T -T r≤R/q, (r,q)=1 μ 2 (r) ϕ(r) mn∼X (mn,q)=1, m>M 0 Λ(m)v r (n) (mn) it e mna q dt M,N ⎛ ⎝ G q R q r≤R/q μ 2 (r) ϕ(r) ⎛ ⎝ T -T S r a q , t, β, M, N dt ⎞ ⎠ 2 ⎞ ⎠ 1/2
. Now we release the condition mn ∼ X:

⎛ ⎝ T -T S r a q , t, β, M, N dt ⎞ ⎠ 2 δT M n∼N |v r (n)| 2 + Δ -Δ T -T b mod * q m≡b[q], m∼M Λ(m) m i(v+t) 2 |H (v)| dt dv × Δ -Δ T -T b mod * q (n,q)=1, n∼N v r (n) n i(v+t) e mna q 2 dt |H (v)| dv. (9.1)
We use the estimate

Δ -Δ T -T v+t=w |H (v)| dv log min(2 + T, Δ),
and, from this point onwards, the treatment of the factor containing v r (n) is as previously. As for the factor containing the Λ-part, we first reduce it to prime variables and then use multiplicative characters. The first step reads

b mod * q m≡b[q], m∼M Λ(m) m iw 2 = b mod * q p≡b[q], p∼M log p p iw + m≡b[q], m∼M m=p k , k≥2 Λ(m) m iw 2 ≤ 2 b mod * q p≡b[q], p∼M log p p iw 2 + 2 b mod * q m≡b[q], m∼M m=p k , k≥2 Λ(m) m iw 2 .
We then use

b mod * q m≡b[q], m∼M m=p k , k≥2 Λ(m) m iw 2 ≤ max b mod * q ⎛ ⎜ ⎝ m≡b[q], m∼M m=p k , k≥2 Λ(m) ⎞ ⎟ ⎠ m∼M m=p k , k≥2 Λ(m) ≤ ⎛ ⎜ ⎝ m≤M m=p k , k≥2 Λ(m) ⎞ ⎟ ⎠ 2 M.
On detecting the congruence condition in m ≡ b[q] by multiplicative characters, we thus find that b mod * q m≡b[q], m∼M

Λ(m) m iw 2 ≤ 2 ϕ(q) χ mod q (p,q)=1, p∼M χ(p) log p p iw 2 + O(M ).
The reduction to primitive characters is immediate:

χ mod q (p,q)=1, p∼M χ(p) log p p iw 2 = f|q χ mod * f (p,q)=1, p∼M χ(p) log p p iw 2 .
The next step is a way to introduce the Barban-Davenport-Halberstam theorem via Theorem 9.

Here f | q, and we note that c r (p) = μ(p) = -1:

r≤M/f, (r,f)=1 μ 2 (r) ϕ(r) (p,q)=1, p∼M χ(p)c r (p) log p p iw 2 = G f M f (p,q)=1, p∼M χ(p) log p p iw 2 .
We recall (3.7):

(f/ϕ(f))G f (M/f) ≥ G(M/f).
We can then appeal to Theorem 9 instead of Lemma 16.

Let us summarize (for uniformity with the last section) the diverse conditions: for some > 0,

(1) R ≥ 8q;

(2) M 0 , R, z ≥ X ;

(3) δM 0 ≥ M 0 for the separation of variables;

(4) δ 2 T R log 6 R ≤ 1 for the large sieve argument on the remainder term coming from the separation of variables, since we integrate trivially in t the corresponding error term from Section 8;

(5) Δ = 100/δ 2 since β = 0, and this makes condition (5) from Section 8 useless here;

(6) T R 2 /(δ 2 z) ≤ 1 and M 0 ≥ q 1+ for the large sieve argument on the bilinear form;

(7) qT 2 Rz 2 ≤ X √ q for the remainder term linked to L

(1) r (a, t, 0); (8) qT 2 Rz 2 M 0 ≤ X √ q for the remainder term linked to L

(2) r (a, t, 0). Condition ( 7) is yet again a consequence of condition [START_REF] Gallagher | A large sieve density estimate near σ = 1[END_REF]. Here are the parameters we choose, for some θ > 0:

R = qX 2θ , δ -1 = qT X 2θ , M 0 = qT X 3θ , z = q 3 T 2 X 4θ , ( 9.2)
and assume that T 13/2 q 8 X 13θ ≤ X. We set η = (1 -13θ)/8 to get the claimed estimate.

THE SHORT INTERVAL ESTIMATE: PROOF OF THEOREM 3

The L 1 -estimate opens a clear path to the short interval result. Let us start with a methodological comment. One can try to compute the Mellin transform of the characteristic function of the interval [X, X + X θ ], but the lack of continuity results in a transform which is of order 1/s when s goes to i∞. As an implication, the integral of the absolute value of this transform over a vertical line is not convergent, and this raises complications. One can get a more precise version for the characteristic function of [1, X] that does not rely on the absolute value of the Mellin transform but relies on more information on the sequence we consider; this is the truncated Perron's formula. Alternatively, one can consider a smooth sum of the initial characteristic functions; then removing the smoothing depends on short interval estimates, and this is exactly the same information that is required in the truncated Perron's formula.

This means that we get results of similar strength by using the difference of two truncated Perron's formulas. A usual form of this truncated Perron's formula like [27, Theorem 2.1] is enough. We note on the methodological side that the more powerful version [33, Theorem 1.2] by the first author would lead to a refined error term.

Theorem 42 (truncated Perron's formula). Let F (z) = n u n /n z be a Dirichlet series that converges absolutely for Re z > κ a , and let κ > 0 be strictly larger than κ a . For X ≥ 1 and T ≥ 1, we have

n≤X u n = 1 2iπ κ+iT κ-iT F (z) X z dz z + O * ⎛ ⎝ ∞ 1/T |log(X/n)|≤v |u n | n κ 2X κ dv T v 2 ⎞ ⎠ .
We take F (z) = X< ≤2X Λ( )e( β)/ z . We select κ = 1 and set ω = X θ-1 . To handle the error term, we split the integral at v = 1. When v ≥ 1, we majorize |log(X/n)|≤v |u n |/n by n∼X Λ(n)/n 1. When v < 1, we majorize 1/n by X -1 e 1/T 1/X. We assume that T ≤ √ X. This implies that, when 1/T ≤ v ≤ 1, we have

|log(X/n)|≤v |u n | n X -1 e -v X≤n≤e v X Λ(n) X -1 vX log(3vX) log X v
by using Lemma 14 with q = 1, A = e -v X and B = 3vX ≥ 2 sinh(v)X ≥ 2X(e ve -v )/2. This readily leads to

X< ≤X+X θ Λ( ) e a q = 1 2iπ 1+iT 1-iT X< ≤2X Λ( )e( a/q) z X z ((1 + ω) z -1) dz z + O X log X T . (10.1)
A usage of [33, Theorem 1.2] we mentioned would remove the log X in the error term, the only slight bump in the proof being that we need the same T * for the formula up to X and the formula up to X + X θ : going back to [33, Corollary 5.1] would do the trick. We end this methodological remark here and proceed by using the classical inequality

|(1 + ω) z -1| ≤ |zω(1 + ω)| obtained from the representation (1 + ω) z -1 = z ω 0 (1 + y) z-1 dy. We choose T = X 1-θ √ q log X.
We shift the line of integration in (10.1) to Re z = 0. To bound the contribution of the two horizontal segments, we notice that, when Im z = ±T and 0

≤ Re z = σ ≤ 1, we have |(1 + ω) z -1| 1, |1/z| 1/T and ∼X Λ( )/| z | X/X σ .
This gives the error term O(X/T ), which is admissible. On the line Re z = 0, we use the estimate

|(1 + ω) z -1|/|z| X θ-1 , getting X< ≤X+X θ Λ( ) e a q X θ-1 T -T X< ≤2X Λ( ) e( a/q) it dt + X θ √ q . ( 10.2)
We are ready to prove Theorem 3. To meet its hypotheses, we require

X 1-θ √ q log X ≤ X η q 16/13 . ( 10.3)
The hypotheses θ > θ 0 and X 1-θ 0 √ q ≤ X η q 16/13 (10.4) are enough to ensure (10.3), when X is large enough.

THE TRIGONOMETRIC POLYNOMIAL OF THE M ÖBIUS FUNCTION

Adapting the previous argument to the polynomial S a q = ∼X μ( ) e a q (11.1) is easy enough. We take the same notation as for the von Mangoldt function, but we add the superscript when it concerns the Möbius function. The treatment of the two linear forms carries over with few changes and same result. The treatment of the bilinear form is simpler in one aspect, since we do not need any Brun-Titchmarsh theorem and use instead of Lemma 17 the estimate

b mod q m≡b[q], m∼M μ(m) 2 M 2 q (11.2)
valid when M ≥ q 1+ . We also consider the companion

S a q = ∼X, ( ,q)=1 μ( ) e a q . ( 11.3)
We have to bound in (7.3)

b mod * q m≡b[q], m∼M μ(m) 2 . (11.4)
This quantity is at most M 2 ϕ(q)/q 2 , while we used M 2 /ϕ(q) when dealing with primes. We thus multiply this bound by ϕ(q) 2 /q 2 , and we have to take the square root of this factor when modifying the final bound.

12. PROOF OF THEOREM 1

We are now in a position to prove Theorem 1. We consider only the case of irregular numbers. Here is a more general statement.

Theorem 43. Let X ≥ 3 and θ ∈ [0.79, 1] be two parameters. Let S be the set of irregular numbers in [X, X + X θ ]. Let q ≤ X 1/20 be a prime, and let A and B be two arbitrary sets in Z/qZ such that |A| • |B| ≥ q(log q) 2 . We have a+b+s≡m[q] a∈A, b∈B, s∈S

∼ |A| • |B| • |S| q

(as q goes to infinity) valid for every m ∈ Z/qZ.

Proof. Let us define two trigonometric polynomials

T A, c q = a∈A e ac q , T B, c q = b∈B e bc q , ( 12.1)
then one over the irregular numbers

U (α) = X≤s≤X+X θ 1 -λ(s) 2 e(sα) (12.2)
and lastly the number of representations

r(A, B, m) = a+b+s≡m[q] a∈A, b∈B, s∈S 1. (12.3)
We get classically

r(A, B, m) = 1 q c mod * q T A, c q T B, c q U c q e - mc q . ( 12.4)
By Theorem 3 with η = 1/12 and λ instead of μ (i.e., in version (V2)), together with the fact that q is prime, we infer that, when c = 0, we have U c q X θ log q ϕ(q) . Furthermore, U (0) ∼ X θ /2 by the result [START_REF] Ramachandra | Some problems of analytic number theory[END_REF] of K. Ramachandra (since θ > 7/12). Hence, by applying Parseval's equality and since ϕ(q) = q -1, we obtain

r(A, B, m) = (1 + o(1)) X θ |A| • |B| 2q + O X θ |A| • |B| log q q .
This proves our claim.

Final note. We have decided to select specific parameters in Theorem 1 to illustrate the relative strength of our result; a larger domain for the parameters (θ, log q/log X) is available.

Proof of Theorem 1. We simply need to select X = q 20r and X θ = q 16r in Theorem 43.

Appendix. SEPARATING THE VARIABLES AND REMOVING A PHASE In the bilinear form, we will have to handle conditions like

a ∼ A, b ∼ B and ab ∼ X,
where y ∼ Y means that Y < y ≤ 2Y . The last condition is annoying because it links both variables together. This link is mild, and we remove it by the process we describe here. At the same time we remove a phase e(βr). Our object here is for some general (ϕ ) and a parameter b > 1 that is usually equal to 2. The same process is used, for instance, in [7, Lemma 6]. Our form is more precise in two aspects: we regulate the length of the integration, and we avoid a loss of a log-factor by introducing (or, more precisely, preparing the introduction of) some information on the local behavior of (|ϕ r |). We rewrite the above as The following L 1 -bounds also hold : The separation of variables relies on the next lemma.

Lemma 45. Let b > 1, δ ∈ (0, 1/2), β and X ≥ 1 be four real parameters. There exists a C 1 -function H such that, for any sequence (ϕ r ) of complex numbers, we have The function H depends on X except when β = 0, so we could dispense with the factor X iu , but it is more natural to keep it. We have in fact On selecting b = 2, using 2 δ/2 -1 ≤ δ and 1 -2 -δ/2 ≤ δ/2, simplifying the constant and using δ -1 + 40|β|X + 60(βX) 2 ≤ 100(δ -1 + (βX) 2 ), we obtain the following result.

Lemma 46. Let δ ∈ (0, 1/2), β and X ≥ 1 be three real parameters. There exists a C 1 -function H such that, for any sequence (ϕ r ) of complex numbers, we have

ACKNOWLEDGMENTS

The gist of this paper emerged while the first author was giving a course on local models and the Hoheisel theorem at IMSc Chennai in 2010. This paper was finalized when both authors were visiting this institute in October 2018. We thank this institute for the pleasant working conditions it offered to us.

Both authors express warm thanks to the referee for his/her very careful reading of this paper.

  of these Ramanujan sums c r are the functions v r defined by v r (m) = c r (m)

( 1 )

 1 r (a, t, β) by Lemma 35; (8) (|t| + 1 + |β|X)

( 1 πv πv 2 .δ when 1 -. 3 )

 1213 r≥1 ϕ r ½ [-1,1] 2 log(r/X) log b -1 e λe κ(2 log(r/X)/log b-1) (A.2) with λ = βX √ b and κ = (log b)/2 and we first seek an approximation for ½ [-1,1] . Let δ ∈ (0, 1/2]be a real parameter. We first note that∞ -∞ -|u|) + e(uv) du = sinWe then consider the trapezoid functionh 0 (δ; u) = (1 -|u|) + -(1δ -|u|) δ ≤ |u| ≤ 1, 0 when 1 ≤ |u|.This function satisfiesh 0 (δ; v) = ∞ -∞ h 0 (δ; u)e(uv) du = (sin πv) 2 -(sin π(1δ)v) 2 π 2 δv 2 = sin(πδv) sin(π(2δ)v) π 2 δv 2 .We further select two additional real parameters κ and λ and buildH(δ,λ, κ; u) = h 0 (δ; u)e(λe κu ). (ALemma 44. Except when u ∈ {±1, ±(1δ)}, we have |H(δ, λ, κ; u)| ≤ 1, |H (δ, λ, κ; u)| ≤ δ -1 + 2π|λκ|e |κ| .

1 - 1 |H 1 - 1 |H

 1111 (δ, λ, κ; u)| du ≤ 2 + 4π|λκ|e |κ| , (δ, λ, κ; u)| du ≤ 4π|λκ|e |κ| 2 + |κ| + |2πλκ|e |κ| . Proof. When u / ∈ {±1, ±(1δ)}, we have H (δ, λ, κ; u) = h 0 (δ; u)e(λe κu ) + 2iπλκe κu H(δ, λ, κ; u) and H (δ, λ, κ; u) 2iπλκe κu = 2h 0 (δ; u)e(λe κu ) + 1 + 2iπλκe κu H(δ, λ, κ; u). The reader will easily derive the lemma from these two expressions. The Fourier transform being defined by f (v) = R f (u)e(uv) du, we deduce from the above lemma and one and then two integrations by parts that | H(δ, λ, κ; v)| ≤ 1 + 2π|λκ|e |κ| π|u| (A.4) and | H(δ, λ, κ; v)| ≤ δ -1 2π 2 |v| 2 + 3 + |κ| + |2πλκ|e |κ| π|v| 2 |λκ|e |κ| . (A.5)

6 )

 6 X<r≤bXϕ r e(βr)= Δ -Δ r≥1 ϕ r r iu H (u)X iu du + O * X<r≤b δ/2 X |ϕ r | + b 1-δ/2 X<r≤bX |ϕ r | + 2δ r≥1 |ϕ r |where Δ is given byΔ = 2 π(log b)δ 2 + 12b + 2b log b + 4πb 2 (log b) |β|X log bAny choice of Δ larger than this one would also do. We have furthermore|H (u)| ≤ (log b)(1 + 2πb|β|X) 4π 2 |u| and ∞ -∞ |H (u)| 2 du = (log b) 2 (2 -2δ) (4π) 2 .

H- 1 + 1 -Δ 1 H

 111 (u) = log b 4π H δ, λ, κ; u log b 4π b iu/2 with λ = βX √ b and κ = log b 2 . (A.7)Proof. The first step is to introduce H(δ, λ, κ; u) (with the parameters we have just specified):X<r≤bX ϕ r e(βr) = r≥1 ϕ r H δ, λ, κ; 2 log(r/X) log b O * 0≤log(r/X)/log b≤δ/2 |ϕ r | + 1-δ/2≤log(r/X)/log b≤1 |ϕ r | .Next, we write (A.5) in the form | H(δ, λ, κ; v)| ≤ (δΔ 1 )/|v| 2 withΔ 1 = 1 2π 2 δ 2 + 6b + b log b + 2πb 2 (log b) |β|X 2π |β|X δ .We truncate the integral and infer thatH δ, λ, κ; 2 log( (δ, λ, κ; v) L 4iπv/log b e(v) dv + O * (2δ).The change of variable u = 4πv/log b concludes. The value of ∞ -∞ |H (u)| 2 du is obtained by appealing to Parseval's equality.

X<r≤2Xϕ 1 + |u| 2 .

 12 r e(βr)= Δ -Δ r≥1 ϕ r r iu H (u)X iu du + O * ⎛ ⎜ ⎝ X<r≤(1+δ)X or (2-δ)X<r≤2X |ϕ r | + 2δ r≥1 |ϕ r | ⎞ ⎟ ⎠where Δ = 100(δ -1 + (βX) 2 )/δ. We have furthermore|H (u)| ≤ 25 73 1 + |β|X 1 + |u| and |H (u)| ≤ δ -1 + (βX) 2Concerning the last bound, we prove in fact that |H (u)| ≤ min log infer the bound stated (we use the first value when |u| ≤ 2).

The surname Vehov is also sometimes spelled Vekhov, in accordance with the traditional transliteration rules for the Cyrillic alphabet.PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICSVol. 314

PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICSVol. 314

PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICSVol. 314 2021

FUNDING

The first author was supported by the CEFIPRA project 5401-A, and the second author was supported by the SERB project SERB/ECR/2018/000850.