who helped me so much when I Wei Zhang

China Thank

Ruihua Wang

In Netherlands

Bin Chen

In Belgium

Mengdi Wang

In Sweden

Chi Cui

Qiyu Yang

Macao Thank

Jie Hu

Li Li

Bin Liu

Long Liu

Jianxi Mao

Baojun Wu

Luming Zhao

Thank

Edouard Daviaud

Jing Feng

Yubin He

Xuanying Li

Yuhui Liu

Quanwu Mu

Lei Shang

Kunkun Song

Jin Tan

Shan Wang

Tingting Wen

Wanlou Wu

Qian Zhang

I would like to thank my brother and my parents for their warmth and caring. At last, I want to express my heartfelt gratitude to Mengyu Tang, who has given me constant encouragement and lasting companionship

I would like to thank my supervisor Professor Jie Wu. It was in 2015 when I first met him. He was giving a report on friable numbers. I was deeply impressed by his erudite knowledge and lively lectures. He is not only a talented mathematician but also an approachable teacher. He loves life and enjoys life. He exercises every day. There is always a smile on his face. He always tells us don't be afraid of making mistakes when doing research. It's him who brought me to the subject of the Riemann zeta function. He has spent plenty of time discussing with me, as well as teaching me how to write a beautiful article. I am so fortunate to be a Ph.D student under his guidance. I have learned so much from him, not only about doing mathematics. Without his guidance, this thesis would not come into appearance.

I want to thank the members of my committee: Professor Christoph Aistleitner, Professor Gautami Bhowmik, Professor Emmanuel Royer and Professor Youness Lamzouri. All of them have carefully read this thesis and made valuable comments and suggestions. Professor Aistleitner encouraged me so much and helped me with job applications. Professor Lamzouri spent a large amount of time helping me with revising this thesis.

Thank Professor Kristian Seip who invited me to attend the conference in Norway. We had a nice discussion. It was also an excellent opportunity to get to know many professors including Andriy Bondarenko, Emanuel Carneiro, Adam Harper and Xu Wang. Thank all of them. I want to thank Professor Leo Goldmakher for helping me find an important article. Thank Marc Munsch for sharing me their relating work on extreme values of Dirichlet Lfunctions. Thank Masahiro Mine for discussions on the distribution of values of the Riemann zeta function. Thank Junxian Li and Jing Zhao for discussions on large values of the Riemann zeta function and thank Daodao Yang for discussions on large values of the derivatives of the Riemann zeta function.

I want to thank Professor Lingmin Liao. He helped me so much with my life in France. I also thank all the professors in our laboratory LAMA, and thank our secretary Sonia Boufala for enthusiastic help during my Ph.D studies in LAMA.

I want to thank my master's advisor Professor Xiumin Ren. During my graduate study in Shandong University, she taught me a lot and helped me so much. I'm also grateful to all the professors of the Number Theory Group in SDU, including: Bingrong Hunag, Jianya Liu, Yujiao Jiang, Guangshi Lü, Xianchang Meng, Zhao Xu, Lilu Zhao and Shuai Zhai. I also want to thank Professor Yong Huang in Changsha, Professor Hongze Li in Shanghai, Professor Qiang Wu in Chongqing, Professor Ping Xi in Xi'an and Professor Wenguang Zhai in Beijing.

I have special thanks to my friends Cheng Zhao, Weijia Wang and Yanbin Zhang. We are old friends for over ten years. Math brings us together and our relationship makes us love math more deeply. Thanks to my friend Ben Liu, who helped me with much technical difficulty in mathematics. Thanks also go to my long-time friends Ruhong Cui, Haibo Li, Nian Liu, Baoyou Qu and Jianwei Tan.

I want to thank my friend and collaborator Bin Wei, and my friend Bin Feng. They helped me both in France and China. Thank my friends Xiaoguang He, Jinjiang Li, Miao Lou, Ke Wang, Yuchao Wang, Zhiwei Wang, Xuanxuan Xiao, Boqing Xue, Min Zhang and i Résumé -Abstract Résumé L'étude de la distribution des valeurs de la fonction zêta de Riemann ζ(s) peut remonter au début du XXe siècle lorsque Bohr a montré que pour tout z ∈ C * et ε > 0, il existe une infinité de s avec 1 < e s < 1 + ε telle que ζ(s) = z. Plus tard, en 1932, Bohr et Jessen [START_REF] Bohr | Über die Werteverteilung der Riemannnschen Zetafunktion[END_REF] montrèrent que log ζ(σ + it) a une distribution continue sur le plan complexe pour tout σ > 1 2 . Sur la ligne critique e s = 1 2 , le théorème de la limite centrale de Selberg [START_REF] Selberg | Contributions to the theory of the Riemann zeta-function[END_REF][START_REF] Selberg | Old and new conjectures and results about a class of Dirichlet series[END_REF] indique que log |ζ( 12 + it)| se comporte comme une variable aléatoire Gaussienne de moyenne 0 et de variance 1 2 log 2 T quand T → ∞, où t varie dans [T, 2T ]. Sur la ligne 1 -le bord droite de la bande cririque, Granville et Soundararajan [START_REF] Granville | The Riemann Zeta Functionn and Related Themes: Papers in Honour of Professor K. Ramachandra[END_REF] ont étudié la distribution de |ζ(1 + it)|, qui est asymptotiquement une fonction double-exponentielle. Ensuite, Lamzouri [START_REF] Lamzouri | On the distribution of extreme values of Zeta and L-functions in the stip 1 2 < σ < 1[END_REF] a considéré la distribution de log |ζ(σ + it)| avec 1 2 < σ < 1 fixe et a également obtenu la fonction de distribution asymptotique. Dans cette thèse, basée sur les résultats établis séparément par Granville et Soundararajan [START_REF] Granville | The Riemann Zeta Functionn and Related Themes: Papers in Honour of Professor K. Ramachandra[END_REF], et par Lamzouri [START_REF] Lamzouri | On the distribution of extreme values of Zeta and L-functions in the stip 1 2 < σ < 1[END_REF], nous obtenons des développements d'ordre supérieur des exposants de ces deux fonctions de distribution.

Le problème d'obtenir des grandes valeurs de |ζ( 1 2 + it)| a d'abord été étudié par Titchmarsh [START_REF] Titchmarsh | The theory of the Riemann zeta-function, 2nd edn[END_REF]Theorem 8.12], qui a montré qu'il existe un t arbitrairement grand tel que |ζ( 1 2 + it)| exp((log t) α ) pour tout α < 1 2 . Le meilleur résultat actuel est dû à de la Bretèche et Tenenbaum [START_REF] De La Bretèche | Sommes de Gál et applications[END_REF], qui ont prouvé qu'il existe un t arbitrairement grand tel que |ζ( 1 2 + it)| exp( 2 log t log 3 t/ log 2 t). Cette borne peut être encore loin de la vraie valeur maximale, en tenant compte que Farmer, Gonek et Hugh [START_REF] Farmer | The maximum size of L-functions[END_REF] ont conjecturé que le maximum devrait être exp(

√ 2 2
log t log 2 t). Outre la ligne critique, il est également intéressant d'étudier les grandes valeurs de |ζ(s)| sur la ligne 1 et dans la bande 1 2 < e s < 1. Sur la ligne 1, l'étude peut remonter à 1925 lorsque Littlewood [START_REF] Littlewood | On the Riemann zeta-function[END_REF] a montré qu'il existe un t arbitrairement grand tel que |ζ(1 + it)| (1 + o( 1))e γ log 2 t. Le meilleur résultat actuel est dû à Astleitner, Mahatab et Munsch [START_REF] Aistleitner | Extreme values of the Riemann zeta function on the 1-line[END_REF] qui ont prouvé qu'il existe un t arbitrairement grand tel que |ζ(1+it)| e γ (log 2 t+log 3 t+O(1)). Mis à part le terme O(1), leur résultat confirme une conjecture de Granville et Soundararajan [START_REF] Granville | The Riemann Zeta Functionn and Related Themes: Papers in Honour of Professor K. Ramachandra[END_REF], qui est basée sur une analyse de la distribution de |ζ(1 + it)|. Dans cette thèse, nous donnons une constante effective c à la place de O(1) dans l'inégalité d'Astleitner, Mahatab et Munsch, ce qui la rapproche de la conjecture de Granville et Soundararajan.

Soit 1 2 < σ < 1. En 1928, Titchmarsh [START_REF] Titchmarsh | On an inequality satisfied by the zeta-function of Riemann[END_REF] montra pour la première fois que pour tout ε > 0, il existe un t arbitrairement grand tel que log |ζ(σ + it)| (log t) 1-σ-ε . En 1977, Montgomery [START_REF] Montgomery | Extreme values of the Riemann zeta function[END_REF] a montré qu'il existe un t arbitrairement grand tel que log |ζ(σ + it)| ν(σ)(log t) 1-σ /(log 2 t) σ pour une certaine constante ν(σ). Il a également conjecturé que cette valeur est le véritable ordre du maximum de log |ζ(σ + it)| jusqu'à ν(σ). Toutes les améliorations ultérieures pour ce problème se concentrent sur l'obtention de valeurs plus grandes de ν(σ). En 2011, Lamzouri [START_REF] Lamzouri | On the distribution of extreme values of Zeta and L-functions in the stip 1 2 < σ < 1[END_REF] a donné une valeur conjecturale du maximum de ν(σ). En 2018, Bondarenko et Seip [START_REF] Bondarenko | Note on the resonance method for the Riemann zeta function, 50 years with Hardy spaces[END_REF] ont considéré les cas quand σ

Abstract

The study of the value distribution of the Riemann zeta function ζ(s) can date back to the early twentieth century when Bohr showed that for any z ∈ C * and ε > 0, there exists infinitely many values of s with 1 < e s < 1 + ε such that ζ(s) = z. Later in 1932, Bohr and Jessen [START_REF] Bohr | Über die Werteverteilung der Riemannnschen Zetafunktion[END_REF] showed that log ζ(σ + it) has a continuous distribution on the complex plane for any σ > 1 2 . On the critical line, Selberg's Central Limit Theorem [START_REF] Selberg | Contributions to the theory of the Riemann zeta-function[END_REF][START_REF] Selberg | Old and new conjectures and results about a class of Dirichlet series[END_REF] states that log |ζ( 1 2 + it)| behaves like a complex Gaussian random variable with mean 0 and variance 1 2 log 2 T as T → ∞, where t varies in [T, 2T ]. On the 1-line, Granville and Soundararajan [START_REF] Granville | The Riemann Zeta Functionn and Related Themes: Papers in Honour of Professor K. Ramachandra[END_REF] studied the distribution of |ζ(1 + it)|, which is asymptotically a double exponentially decreasing function. In the critical strip 1 2 < e s < 1, Lamzouri [START_REF] Lamzouri | On the distribution of extreme values of Zeta and L-functions in the stip 1 2 < σ < 1[END_REF] studied the distribution of log |ζ(σ + it)| with any fixed 1 2 < σ < 1 and also got the asymptotic distribution function. In this thesis, based on the results established separately by Granville and Soundararajan [START_REF] Granville | The Riemann Zeta Functionn and Related Themes: Papers in Honour of Professor K. Ramachandra[END_REF] and Lamzouri [START_REF] Lamzouri | On the distribution of extreme values of Zeta and L-functions in the stip 1 2 < σ < 1[END_REF], we obtain higher order expansions of the exponents of these two distribution functions.

The problem of getting large values of |ζ( 1 2 + it)| was first considered by Titchmarsh [START_REF] Titchmarsh | The theory of the Riemann zeta-function, 2nd edn[END_REF]Theorem 8.12], who showed that there exists arbitrarily large t such that |ζ( 1 2 + it)| exp((log t) α ) for any α < 1 2 . The best-known result up to now is due to de la Bretèche and Tenenbaum [START_REF] De La Bretèche | Sommes de Gál et applications[END_REF], who showed that there exists arbitrarily large t such that |ζ( 1 2 + it)| is as large as exp( 2 log t log 3 t/ log 2 t). This bound may still be far from the true maximal value, considering that Farmer, Gonek and Hugh [START_REF] Farmer | The maximum size of L-functions[END_REF] conjectured that the maximum should be exp(

√ 2 2
log t log 2 t). Besides the critical line, it is also interesting to study large values of |ζ(s)| on the 1-line and in the critical strip 1 2 < e s < 1. On the 1-line, it can date back to 1925 when Littlewood [START_REF] Littlewood | On the Riemann zeta-function[END_REF] showed that there exists arbitrarily large t such that |ζ(1 + it)| (1 + o( 1))e γ log 2 t. The best-known result up to now is due to Astleitner, Mahatab and Munsch [START_REF] Aistleitner | Extreme values of the Riemann zeta function on the 1-line[END_REF] who showed that there exists arbitrarily large t such that |ζ(1+it)| is as large as e γ (log 2 t+log 3 t+O(1)). Their result coincides with (up to an error term O(1)) a conjecture of Granville and Soundararajan [START_REF] Granville | The Riemann Zeta Functionn and Related Themes: Papers in Honour of Professor K. Ramachandra[END_REF], which is based on the analysis of the distribution of |ζ(1 + it)|. In this thesis, we give an effective constant c instead of the O(1) in the inequality of Astleitner, Mahatab and Munsch, which makes it closer to the conjecture of Granville and Soundararajan.

Let 1 2 < σ < 1. In 1928 Titchmarsh [START_REF] Titchmarsh | On an inequality satisfied by the zeta-function of Riemann[END_REF] first showed that for any ε > 0 there exists arbitrarily large t such that log |ζ(σ + it)| (log t) 1-σ-ε . In 1977, Montgomery [START_REF] Montgomery | Extreme values of the Riemann zeta function[END_REF] showed that there exists arbitrarily large t such that log |ζ(σ + it)| can be larger than ν(σ)(log t) 1-σ /(log 2 t) σ for some constant ν(σ). He also conjectured that this value is the true order of the maximum of log |ζ(σ + it)| up to ν(σ). All the later improvements for this problem focus on getting larger values of ν(σ). In 2011, Lamzouri [START_REF] Lamzouri | On the distribution of extreme values of Zeta and L-functions in the stip 1 2 < σ < 1[END_REF] gave a conjectural largest value of ν(σ). In 2018, Bondarenko and Seip [START_REF] Bondarenko | Note on the resonance method for the Riemann zeta function, 50 years with Hardy spaces[END_REF] considered the cases when σ L'étude de la distribution des valeurs de ζ(s) dans la bande critique 0 < e s 1 a une longue histoire. Sur la ligne critique e s = 1 2 , nous avons le théorème de la limite centrale de Selberg. Sur la ligne 1 -le bord droite de la bande critique, Granville et Soundararajan [START_REF] Granville | The Riemann Zeta Functionn and Related Themes: Papers in Honour of Professor K. Ramachandra[END_REF] ont étudié la distribution de |ζ(1 + it)|, qui est asymptotiquement une fonction double exponentiellement décroissante . Pour la bande 1 2 < e s < 1, Lamzouri [START_REF] Lamzouri | On the distribution of extreme values of Zeta and L-functions in the stip 1 2 < σ < 1[END_REF] a étudié la distribution de log |ζ(σ + it)| avec tout 1 2 < σ < 1 fixe et a également obtenu la fonction de distribution asymptotique.

En outre, il est également important d'étudier les valeurs extrêmes de la fonction zêta de Riemann. Sur la ligne 1, le résultat très récent établi par Astleitner, Mahatab et Munsch [START_REF] Aistleitner | Extreme values of the Riemann zeta function on the 1-line[END_REF] dit qu'il existe un t arbitrairement grand tel que |ζ(1 + it)| puisse être aussi grand que e γ (log 2 t + log 3 t + O(1)). Mis à part le terme O(1), leur résultat confirme une conjecture de Granville et Soundararajan [START_REF] Granville | The Riemann Zeta Functionn and Related Themes: Papers in Honour of Professor K. Ramachandra[END_REF], qui est basée sur une analyse de la distribution de |ζ(1+it)|. Dans la bande 1 2 < e s < 1, Montgomery [START_REF] Montgomery | Extreme values of the Riemann zeta function[END_REF] a prouvé qu'il existe un t arbitrairement grand tel que log |ζ(σ + it)| (c m (σ) + o( 1))(log t) 1-σ /(log t) σ pour une certaine constante c m (σ) dépendant de σ. De plus, Bondarenko et Seip [START_REF] Bondarenko | Note on the resonance method for the Riemann zeta function, 50 years with Hardy spaces[END_REF] ont considéré le cas quand σ

|ζ( 1 2 + it)| exp ( √ 2 2 + o(1)) log T log 3 T log 2 T
pour T → ∞. Un an après, ils ont amélioré la constant √ 2/2 en 1. Le meilleur résultat actuel sur les grandes valeurs de |ζ( 12 + it)| est dû à de la Bretèche et Tenenbaum [START_REF] De La Bretèche | Sommes de Gál et applications[END_REF] :

max t∈[0,T ] |ζ( 1 2 + it)| exp ( √ 2 + o(1)) log T log 3 T log 2 T
.

Notons qu'on obtient au mieux ce résultat en appliquant la méthode des sommes GCD. Le théorème de la limite centrale de Selberg [START_REF] Selberg | Contributions to the theory of the Riemann zeta-function[END_REF][START_REF] Selberg | Old and new conjectures and results about a class of Dirichlet series[END_REF] énonce que quand T → ∞, on a

1 T meas t ∈ [T, 2T ] : log |ζ( 1 2 + it)| 1 2 log 2 T τ → 1 √ 2π ∞ τ e -u 2 du.
En traitant les maxima locaux de log |ζ( 1 2 +it)| comme des variables aléatoires indépendantes satisfaisant le théorème de la limite centrale de Selberg, en 2007, Farmer, Gonek et Hugh [START_REF] Farmer | The maximum size of L-functions[END_REF]

ont conjecturé que max t∈[0,T ] |ζ( 1 2 + it)| = exp ( √ 2 2 + o(1)) log T log 2 T .
Pour la borne supérieure de |ζ(

Φ T (τ ) := 1 T meas t ∈ [T, 2T ] : |ζ(1 + it)| > e γ τ .
De plus, posons

C j := 2 0 log t 2 j log I 0 (t) t 2 dt + ∞ 2 log t 2 j log I 0 (t) -t t 2 dt (0.2.1) avec I 0 (t) := ∞ n=0 (t/2) 2n (n!) 2 •
Ils ont montré que la formule asymptotique

Φ T (τ ) = exp - 2e τ -C 0 -1 τ 1 + O 1 √ τ + e τ log T (0.2.2) a lieu uniformément pour 1 τ < log 2 T -20.
En se basant sur (0.2.2), Granville et Soundararajan [START_REF] Granville | The Riemann Zeta Functionn and Related Themes: Papers in Honour of Professor K. Ramachandra[END_REF]

ont conjecturé que max t∈[T,2T ] |ζ(1 + it)| = e γ (log 2 T + log 3 T + C 0 + 1 -log 2 + o(1)). (0.2.3)
La méthode pour prouver (0.2.2) a été adapté pour étudier la distribution des valeurs d'autres fonctions L sur la ligne 1. Toujours dans [START_REF] Granville | The Riemann Zeta Functionn and Related Themes: Papers in Honour of Professor K. Ramachandra[END_REF], Granville et Soundararajan ont montré que la distribution des fonctions L de Dirichlet à l'aspect de caractères a la même forme que (0.2.2). Ce résultat peut être utilisé pour étudier la distribution des grandes sommes de caractères, voir [START_REF] Bober | The distribution of the maximum of character sums[END_REF] et [START_REF] Bober | The frequency and the structure of large character sums[END_REF]. En 2003, Granville et Soundararajan [START_REF] Granville | The distribution of values of L(1, χ d )[END_REF] ont étudié la distribution des fonctions L de Dirichlet de caractères quadratiques L(1, χ d ), et résolu une conjecture de Montgomery et Vaughan [START_REF] Montgomery | Extreme values of the Riemann zeta function[END_REF]. En 2007, Wu [START_REF] Wu | Note on a paper by A[END_REF] a amélioré ce résultat de [START_REF] Granville | The distribution of values of L(1, χ d )[END_REF] en donnant un développement d'ordre supérieur dans l'exposant de la fonction de distribution. En 2008, Liu, Royer et Wu [START_REF] Liu | On a conjecture of Montgomery-Vaughan on extreme values of automorphic L-functions at 1, Anatomy of integers[END_REF] ont considéré le cas de fonctions L de puissance symétriques automorphes. En 2010, Lamzouri [START_REF] Lamzouri | Distribution of values of L-functions at the edge of the critical strip[END_REF] a étudié des fonctions L générales qui peuvent couvrir les résultats de [START_REF] Granville | The distribution of values of L(1, χ d )[END_REF][START_REF] Liu | On a conjecture of Montgomery-Vaughan on extreme values of automorphic L-functions at 1, Anatomy of integers[END_REF]. En 2008, Lamzouri [START_REF] Lamzouri | The two dimensional distribution of values of ζ(1 + it)[END_REF] généralise (0.2.2) à la distribution conjointe de arg

ζ(1 + it) et |ζ(1 + it)|.
Inspirés par le résultat de Wu [START_REF] Wu | Note on a paper by A[END_REF], nous cherchons à obtenir une amélioration de (0.2.2), qui présente une expansion d'ordre supérieur dans l'exposant. Théorème 0.2.1. (Theorem 3.1.1) Il existe une suite de nombres réels {a j } j 1 telle que pour tout entier J 1 nous avons

Φ T (τ ) = exp - 2e τ -C 0 -1 τ 1 + J j=1 a j τ j + O J 1 τ J+1 + e τ log T uniformément pour T → ∞ et 1 τ log 2 T -20, où C 0 est défini comme dans (0.2.1). De plus, a 1 = 2(1 + C 0 -C 1 ).
Récemment, en 2018, Aistleitner, Mahatab et Munsch [START_REF] Aistleitner | Extreme values of the Riemann zeta function on the 1-line[END_REF] ont utilisé la méthode de "résonance longue" pour montrer que max

t∈[ √ T ,T ] |ζ(1 + it)| e γ (log 2 T + log 3 T + O(1)).
Notons que cela nécessite un plus grand intervalle [ √ T , T ] que [T, 2T ] dans (0.2.3) ce qui est typique pour l'application de la méthode de "résonance longue". Inspirés par leur travail, nous établissons le Théorème 0.2.2 ci-dessous contenant une estimation des valeurs extrêmes dans l'intervalle [T β , T ] avec une constante c effective.

Théorème 0.2.2. (Theorem 4.1.1) Soient 0 < β < 1 et c une constante telle que c < log(1 -β) -log 2 4 -1. Définissons Z β (T ) := max T β |t| T |ζ(1 + it)|.
Alors pour T suffisamment grand, nous avons Il est aussi très intéressant d'étudier les valeurs extrêmes des dérivées de la fonction zêta de Riemann. Pour tout ∈ N + , définissons

Z β (T ) e γ (log 2 T + log 3 T + c).

Notons que lorsque

Z ( ) (T ) := max t∈[T,2T ] |ζ ( ) (1 + it)|.
Outre d'autres résultats, Yang [START_REF] Yang | Extreme values of derivatives of the Riemann zeta function[END_REF] a récemment prouvé que nous avons Lamzouri [START_REF] Lamzouri | On the distribution of extreme values of Zeta and L-functions in the stip 1 2 < σ < 1[END_REF] a montré qu'il existe une constante positive c(σ), telle que nous avons

Z ( ) (T ) e γ ( + 1) +1 {log 2 T -log 3 T + O(1)} +1 , (0.2.4) uniformément pour T → ∞ et (log T )/(log 2 T ),

Dans la bande

Φ T (τ, σ) = exp -(τ log σ τ ) 1 1-σ a 0 (σ) + O 1 √ log τ + (τ log τ ) 1 1-σ log T σ-1 2 , (0.3.2) uniformément pour 1 τ c(σ)(log T ) 1-σ / log 2 T , où a 0 (σ) := σ 2 (1 -σ) 1/σ-2 C 0 (σ) σ 1-σ (0.3.3) avec C 0 (σ) := ∞ 0 log I 0 (t) t 1/σ+1 dt (n 0). (0.3.4)
Nous visons à améliorer la fonction de distribution (0.3.1). Nous avons une expansion d'ordre supérieur dans l'exposant, qui est inspirée par le travail dans [START_REF] Wu | Note on a paper by A[END_REF]. Nous nous référons à [START_REF] Mine | Large deviations for values of L-functions attached to cusp forms in the level aspect[END_REF] pour des travaux similaires sur les fonctions L attachées aux formes modulaires cuspidales.

Théorème 0.3.1. (Theorem 3.1.1) Il existe une suite de polynômes à coefficients réels (dépendants de σ) {a n (σ, •)} n 0 avec deg(a n ) n, et une constante c(σ) > 0, telles que pour tout entier N 1, on a

Φ T (τ, σ) = exp -(τ log σ τ ) 1 1-σ N n=0 a n (σ, log 2 τ ) (log τ ) n + O log 2 τ log τ N +1 + (τ log τ ) 1 1-σ log T σ-1 2 uniformément pour T → ∞ et 1 τ c(σ)(log T ) 1-σ / log 2 T . En particulier, nous avons a 0 (σ, •) = a 0 (σ).
Maintenant nous nous tournons vers les grandes valeurs de |ζ(σ + it)|. En 1928, Titchmarsh [START_REF] Titchmarsh | On an inequality satisfied by the zeta-function of Riemann[END_REF] a montré que pour tout ε > 0, nous avons lim sup

t→∞ log |ζ(σ + it)| (log t) 1-σ-ε = ∞.
En 1972, Levinson [START_REF] Levinson | Ω-theorems for the Riemann zeta-function[END_REF] a amélioré ce résultat en montrant que pour T suffisamment grand:

max t∈[0,T ] log |ζ(σ + it)| (log T ) 1-σ log 2 T .
En 1977, Montgomery [START_REF] Montgomery | Extreme values of the Riemann zeta function[END_REF] a montré que max

t∈[0,T ] log |ζ(σ + it)| ν(σ) (log T ) 1-σ (log 2 T ) σ , (0.3.5) où ν(σ) = 1 20 σ -1 2 , et ν(σ) = 1
20 sous l'hypothèse de Riemann. Il est conjecturé que la quantité (log T ) 1-σ /(log 2 T ) σ soit l'ordre de grandeur réel de max t∈[T,2T ] log |ζ(σ + it)|. Ainsi, la seule amélioration possible de (0.3.4) à laquelle nous pourrions nous attendre est d'obtenir des valeurs plus grandes de ν(σ). En tenant compte de (0.3.1), Lamzouri (voir [START_REF] Lamzouri | On the distribution of extreme values of Zeta and L-functions in the stip 1 2 < σ < 1[END_REF]) a proposé la conjecture suivante :

max t∈[T,2T ] log |ζ(σ + it)| = (ν * (σ) + o(1)) (log T ) 1-σ (log 2 T ) σ où ν * (σ) := C 0 (σ)σ -2σ (1 -σ) σ-1 et C 0 (σ)
ν(σ) = ( √ 2 2 + o(1)) | log(σ -1 2 )| σ 1 2 , (1 -σ) -1 + O(| log(1 -σ)|) σ 1, (0.3.6) pour σ ∈ ( 1 2 + 1 log 2 T , 1 -1 log 2 T ). Ici, σ 1
< β < 1. Alors pour T → ∞ et 1 2 + 1 log 2 T < σ < 3 4 , nous avons max t∈[T β ,T ] log |ζ(σ + it)| ν β (σ) (log T ) 1-σ (log 2 T ) σ , où ν β (σ) := ( √ 2 + o(1))(1 -β) 1-σ | log(σ - 1
2 )|. Par conséquent, lorsque β 0, nous pouvons également choisir

ν(σ) = ( √ 2 + o(1)) | log(σ -1 2 )|.

Plan de cette thèse

Le Chapitre 0 est une introduction dans laquelle nous présentons nos résultats principaux obtenus dans cette thèse.

Dans le chapitre 1, nous introduisons quelques connaissances préliminaires, y compris certaines propriétés fondamentales de la fonction zêta de Riemann que nous devons utiliser dans les chapitres suivants.

Les Chapitres 2-5 sont le coeur de cette thèse : Dans le chapitre 2, nous étudions la distribution des grandes valeurs de |ζ(1 + it)|. Dans le chapitre 3, nous étudions la distribution des grandes valeurs de |ζ(σ + it)| dans la bande 1 2 < σ < 1. Dans le chapitre 4 (en collaboration avec Bin Wei), nous étudions séparément les grandes valeurs de la fonction zêta de Riemann sur la ligne 1 et dans la bande 1 2 < e s < 1. Dans le chapitre 5 (en collaboration avec Bin Wei), nous étudions de grandes valeurs des dérivées de la fonction zêta de Riemann sur la ligne 1.

Introduction

The Riemann zeta function plays a pivotal role in analytic number theory, and has applications in physics, probability theory, and applied statistics. The study of the distribution of values of the Riemann zeta function has a long history. On the critical line, we have Selberg's central limit theorem. On the 1-line, Granville and Soundararajan [START_REF] Granville | The Riemann Zeta Functionn and Related Themes: Papers in Honour of Professor K. Ramachandra[END_REF] studied the distribution of |ζ(1+it)|, which is asymptotically a double exponentially decreasing function. In the critical strip 1 2 < e s < 1, Lamzouri [START_REF] Lamzouri | On the distribution of extreme values of Zeta and L-functions in the stip 1 2 < σ < 1[END_REF] studied the distribution of log |ζ(σ + it)| with any fixed 1 2 < σ < 1 and also got the asymptotic distribution function. Besides, it is also important to investigate the extreme values of the Riemann zeta function. On the 1-line, the very recent result established by Astleitner, Mahatab and Munsch [START_REF] Aistleitner | Extreme values of the Riemann zeta function on the 1-line[END_REF] says that there exists arbitrarily large t such that |ζ(1 + it)| can be as large as e γ (log 2 t + log 3 t + O(1)). Their result coincides with a conjecture of Granville and Soundararajan [START_REF] Granville | The Riemann Zeta Functionn and Related Themes: Papers in Honour of Professor K. Ramachandra[END_REF], which is based on the analysis of the distribution of |ζ(1 + it)|. In the critical strip 1 2 < e s < 1, Montgomery [START_REF] Montgomery | Extreme values of the Riemann zeta function[END_REF] proved that there exists arbitrarily large t such that log |ζ(σ + it)| (ν(σ) + o( 1))(log t) 1-σ /(log t) σ for some constant ν(σ) depending on σ. Moreover, Bondarenko and Seip [START_REF] Bondarenko | Note on the resonance method for the Riemann zeta function, 50 years with Hardy spaces[END_REF] considered the case when σ and showed that there exists arbitrarily large t such that log |ζ(σ + it)| can be larger than

( √ 2 2 + o(1)) log |σ -1 2 |(log t) 1-σ /(log t) σ .
In the following three sections, we will introduce these subjects in detail.

On the critical line

Titchmarsh [START_REF] Titchmarsh | The theory of the Riemann zeta-function, 2nd edn[END_REF]Theorem 8.12] showed that there exists arbitrarily large t such that for any α < 1 2 |ζ( 1 2 + it)| exp((log t) α ). In 1977, assuming the Riemann Hypothesis (RH), Montgomery [START_REF] Montgomery | Extreme values of the Riemann zeta function[END_REF] showed that there exists arbitrarily large t such that

|ζ( 1 2 + it)| > exp 1 20 log t log 2 t ,
where log j is the j-th iterated logarithm. In 1977, Balasubramanian and Ramachandra [START_REF] Balasubramanian | On the frequency of Titchmarsh's phenomenon for ζ(s), III[END_REF] showed unconditionally (without RH) there exists arbitrarily large t such that

|ζ( 1 2 + it)| > exp c log t log 2 t ,
where c is a constant. In 1986, Balasubramanian [START_REF] Balasubramanian | On the frequency of Titchmarsh's phenomenon for ζ(s)[END_REF] gave an effective value of this constant: c = 0.530 . . . In 2008, Soundararajan [START_REF] Soundararajan | Extreme values of zeta and L-functions[END_REF] showed by using the resonance method that c = 1 + o(1). In 2017, Bondarenko and Seip [START_REF] Bondarenko | Large greatest common divisor sums and extreme values of the Riemann zeta function[END_REF] made a breakthrough by showing that max

t∈[0,T ] |ζ( 1 2 + it)| exp ( √ 2 2 + o(1)) log T log 3 T log 2 T
, as T → ∞. One year later, they [START_REF] Bondarenko | Extreme values of the Riemann zeta function and its argument[END_REF] improved the constant

√ 2
2 to 1. The best-known result on large values of |ζ( 1 2 + it)| is due to de la Bretèche and Tenenbaum [START_REF] De La Bretèche | Sommes de Gál et applications[END_REF], who in 2018 showed that max

t∈[0,T ] |ζ( 1 2 + it)| exp ( √ 2 + o(1)) log T log 3 T log 2 T
.

This result was established by using the method of GCD sums. Selberg's central limit theorem [START_REF] Selberg | Contributions to the theory of the Riemann zeta-function[END_REF][START_REF] Selberg | Old and new conjectures and results about a class of Dirichlet series[END_REF] says that:

1 T meas t ∈ [T, 2T ] : log |ζ( 1 2 + it)| 1 2 log 2 T τ → 1 √ 2π ∞ τ e -u 2 du,
as T → ∞. Upon treating the local maxima of log |ζ( 1 2 + it)| as statistically independent variables satisfying Selberg's central limit theorem, in 2007 Farmer, Gonek and Hugh [START_REF] Farmer | The maximum size of L-functions[END_REF] conjectured that max

t∈[0,T ] |ζ( 1 2 + it)| = exp{( √ 2 2 + o(1)) log T log 2 T }.
For the upper bound, the Lindelöf hypothesis states that for any ε > 0

ζ( 1 2 + it) ε |t| ε (|t| 1).
The best-known upper bound is due to Bourgain [START_REF] Bourgain | Decoupling, exponential sums and the Riemann zeta function[END_REF] who proved that

ζ( 1 2 + it) ε t 13 84 +ε (|t| 1).
The best conditional bound (assuming the Riemann hypothesis) is the result of Chandee and Soundararajan [START_REF] Chandee | Bounding |ζ( 1 2 + it)| on the Riemann Hypothesis[END_REF] that

ζ( 1 2 + it) exp c log t log 2 t (|t| 3),
for some absolute constant c > 0. For further details, we refer to [START_REF] Balasubramanian | On the frequency of Titchmarsh's phenomenon for ζ(s), III[END_REF][START_REF] Balasubramanian | On the frequency of Titchmarsh's phenomenon for ζ(s)[END_REF][START_REF] Soundararajan | Extreme values of zeta and L-functions[END_REF][START_REF] Titchmarsh | The theory of the Riemann zeta-function, 2nd edn[END_REF].

On the 1-line

In this part, we focus on the values of the Riemann zeta function ζ(s) on the 1-line. The study of the extreme values of |ζ(1 + it)| can date back to 1925 when Littlewood [START_REF] Littlewood | On the Riemann zeta-function[END_REF] showed that there exists arbitrarily large t for which

|ζ(1 + it)| (1 + o(1))e γ log 2 t.
This was improved by Levinson [START_REF] Levinson | Ω-theorems for the Riemann zeta-function[END_REF], who in 1972 proved that there exists arbitrarily large t such that |ζ(1 + it)| e γ log 2 t + O(1).

In 2006, Granville and Soundararajan [START_REF] Granville | The Riemann Zeta Functionn and Related Themes: Papers in Honour of Professor K. Ramachandra[END_REF] used Diophantine approximation to prove that the lower bound

max t∈[1,T ] |ζ(1 + it)| e γ (log 2 T + log 3 T -log 4 T + O(1))
holds for sufficiently large T . In the same article, they also studied the distribution of |ζ(1 + it)|. To be more precise, we first define for T > 1,

Φ T (τ ) := 1 T meas t ∈ [T, 2T ] : |ζ(1 + it)| > e γ τ .
Furthermore, let

C j := 2 0 log t 2 j log I 0 (t) t 2 dt + ∞ 2 log t 2 j log I 0 (t) -t t 2 dt, (0.2.1)
with

I 0 (t) := ∞ n=1 (t/2) 2n (n!) 2 .
Then they showed that

Φ T (τ ) = exp - 2e τ -C 0 -1 τ 1 + O 1 √ τ + e τ log T (0.2.2)
holds uniformly for 1 τ < log 2 T -20. Based on (0.2.2), Granville and Soundararajan [START_REF] Granville | The Riemann Zeta Functionn and Related Themes: Papers in Honour of Professor K. Ramachandra[END_REF]

conjectured that max t∈[T,2T ] |ζ(1 + it)| = e γ (log 2 T + log 3 T + C 0 + 1 -log 2 + o(1)). (0.2.3)
The method to prove (0.2.2) was also adjusted to studying the distribution of values on the 1-line of other L-functions. Also in [START_REF] Granville | The Riemann Zeta Functionn and Related Themes: Papers in Honour of Professor K. Ramachandra[END_REF], Granville and Soundararajan showed that the distribution of the Dirichlet L-functions in the aspect of characters has the same form as (0.2.2). This result can be used to study the distribution of large character sums, see [START_REF] Bober | The distribution of the maximum of character sums[END_REF] and [START_REF] Bober | The frequency and the structure of large character sums[END_REF]. In 2003, Granville and Soundararajan [START_REF] Granville | The distribution of values of L(1, χ d )[END_REF] studied the distribution of the Dirichlet Lfunctions of quadratic characters L(1, χ d ), which proves part of the conjecture of Montgomery and Vaughan in [START_REF] Montgomery | Extreme values of the Riemann zeta function[END_REF]. In 2007, Wu [START_REF] Wu | Note on a paper by A[END_REF] improved this result by giving a higher order expansion in the exponent of the distribution function. In 2008, Liu, Royer and Wu [START_REF] Liu | On a conjecture of Montgomery-Vaughan on extreme values of automorphic L-functions at 1, Anatomy of integers[END_REF] considered the case of symmetric power L-functions. In 2010, Lamzouri [START_REF] Lamzouri | Distribution of values of L-functions at the edge of the critical strip[END_REF] studied the generalized L-functions which can cover the results of [START_REF] Granville | The distribution of values of L(1, χ d )[END_REF][START_REF] Liu | On a conjecture of Montgomery-Vaughan on extreme values of automorphic L-functions at 1, Anatomy of integers[END_REF]. In 2008 Lamzouri [START_REF] Lamzouri | The two dimensional distribution of values of ζ(1 + it)[END_REF] Inspired by the result of Wu [START_REF] Wu | Note on a paper by A[END_REF], we aim to get an improvement of (0.2.2), which presents a higher order expansion in the exponent. Theorem 0.2.1. (Theorem 3.1.1) There is a sequence of real numbers {a j } j 1 such that for any integer J 1 we have

Φ T (τ ) = exp - 2e τ -C 0 -1 τ 1 + J j=1 a j τ j + O J 1 τ J+1 + e τ log T
uniformly for T → ∞ and 1 τ log 2 T -20, where C 0 is defined as in (0.2.1). Moreover,

a 1 = 2(1 + C 0 -C 1 ).
Recently in 2018, Aistleitner, Mahatab and Munsch [START_REF] Aistleitner | Extreme values of the Riemann zeta function on the 1-line[END_REF] used the method of "long resonance" to show that max

t∈[ √ T ,T ] |ζ(1 + it)| e γ (log 2 T + log 3 T + O(1)).
Note that this requires a larger range [ √ T , T ] than [T, 2T ] in (0.2.3) which is typical for the application of "long resonance". Inspired by their work, we establish Theorem 0.2.2 below containing an estimate of extreme values in the range [T β , T ]. Theorem 0.2.2. (Theorem 4.1.1) Let 0 < β < 1 be fixed and c be a constant such that

c < log(1 -β) -log 2 4 -1. Define Z β (T ) = max T β |t| T |ζ(1 + it)|.
Then for sufficiently large T , we have

Z β (T ) e γ (log 2 T + log 3 T + c).
Note that when β = 1 2 , we can choose the constant c = -3.6931472. This gives a description of the error term O(1) in the result by Aistleitner, Mahatab and Munsch. Despite the enlarged range, Theorem 0.2.2 is in accordance with the conjecture (0.2.3) which predicts a larger constant C 0 + 1 -log 2 = -0.0885469.

It also draws wide interests on the extreme values of the derivatives of the Riemann zeta function. For any ∈ N + , denote

Z ( ) (T ) := max t∈[T,2T ] |ζ ( ) (1 + it)|.
Besides other results, Yang [START_REF] Yang | Extreme values of derivatives of the Riemann zeta function[END_REF] recently proved that we have

Z ( ) (T ) e γ ( + 1) +1 {log 2 T -log 3 T + O(1)} +1 , (0.2.4)
uniformly for T → ∞ and (log T )/(log 2 T ). We aim to improve the constant /( +1) +1 in (0.2.4). We have the following theorem.

Theorem 0.2.3. (Theorem 5.1.1) We have

Z ( ) (T ) e γ + 1 (log 2 T ) +1 {1 + o(1)} ,
uniformly for T → ∞ and all positive integers (log T )/(log 2 T ).

0.3 In the strip 1 2 < e s < 1

Let σ ∈ ( 1 2 , 1). In 2011, Lamzouri [START_REF] Lamzouri | On the distribution of extreme values of Zeta and L-functions in the stip 1 2 < σ < 1[END_REF] studied the distribution of large values of |ζ(σ + it)| as t varies in [T, 2T ], by applying a method of Granville and Soundararajan [START_REF] Granville | The Riemann Zeta Functionn and Related Themes: Papers in Honour of Professor K. Ramachandra[END_REF] to investigate the distribution of values of |ζ(1 + it)|. Let T be sufficiently large. Define the distribution function by

Φ T (τ, σ) := 1 T meas{t ∈ [T, 2T ] : log |ζ(σ + it)| > τ }. (0.3.1)
Lamzouri [START_REF] Lamzouri | On the distribution of extreme values of Zeta and L-functions in the stip 1 2 < σ < 1[END_REF] showed that, there exists a positive constant c(σ), such that uniformly in the range 1

τ c(σ)(log T ) 1-σ / log 2 T we have Φ T (τ, σ) = exp -(τ log σ τ ) 1 1-σ a 0 (σ) + O 1 √ log τ + (τ log τ ) 1 1-σ log T σ-1 2 , (0.3.2)
where

a 0 (σ) := σ 2 (1 -σ) 1/σ-2 C 0 (σ) σ 1-σ (0.3.3) with C n (σ) := ∞ 0 (log t) n t 1/σ+1 log I 0 (t) dt (n 0). (0.3.4)
We aim to improve the asymptotic distribution function (0.3.2). We have a higher order expansion in the exponent, which is inspired by the work in [START_REF] Wu | Note on a paper by A[END_REF]. We refer to [START_REF] Mine | Large deviations for values of L-functions attached to cusp forms in the level aspect[END_REF] for similar work on L-functions attached to cusp forms. Theorem 0.3.1. (Theorem 3.1.1) Let Φ T (τ, σ) be defined in (0.3.1). Then there exists a sequence of polynomials with real coefficients {a n (σ, •)} n 0 with deg(a n ) n, and a constant c(σ) > 0, such that for any integer N 1, we have

Φ T (τ, σ) = exp -(τ log σ τ ) 1 1-σ N n=0 a n (σ, log 2 τ ) (log τ ) n + O log 2 τ log τ N +1 + (τ log τ ) 1 1-σ log T σ-1 2 uniformly for T → ∞ and 1 τ c(σ)(log T ) 1-σ / log 2 T . Especially, we have a 0 (σ, •) = a 0 (σ).
Now we turn to the large values of |ζ(σ + it)|. In 1928, Titchmarsh [START_REF] Titchmarsh | On an inequality satisfied by the zeta-function of Riemann[END_REF] showed that for any ε > 0, we have lim sup

t→∞ log |ζ(σ + it)| (log t) 1-σ-ε = ∞.
In 1972, Levinson [34] improved it, by showing that for sufficiently large T we have max

t∈[0,T ] log |ζ(σ + it)| (log T ) 1-σ log 2 T .
In 1977, Montgomery [START_REF] Montgomery | Extreme values of the Riemann zeta function[END_REF] showed that max

t∈[0,T ] log |ζ(σ + it)| ν(σ) (log T ) 1-σ (log 2 T ) σ , (0.3.5)
where ν(σ) = 1 20 σ -1 2 unconditionally, and ν(σ) = 1 20 on assuming the Riemann hypothesis. He also conjectured this quantity (log T ) 1-σ /(log 2 T ) σ is the true order of magnitude of max t∈[T,2T ] log |ζ(σ + it)|. Thus, the only improvement of (0.3.5) we could expect is to get larger values of ν(σ). In 2011, taking (0.3.2) into account, Lamzouri [START_REF] Lamzouri | On the distribution of extreme values of Zeta and L-functions in the stip 1 2 < σ < 1[END_REF]

conjectured : max t∈[T,2T ] log |ζ(σ + it)| = (ν * (σ) + o(1)) (log T ) 1-σ (log 2 T ) σ , where ν * (σ) = C 0 (σ)σ -2σ (1 -σ) σ-1
, and C 0 (σ) is defined in (0.3.4). In 2016, using the resonance method, Aistleitner [START_REF] Aistleitner | Lower bounds for the maximum of the Riemann zeta function along vertical lines[END_REF] improved Montgomery's unconditional result by showing that (0.3.5) holds for ν(σ) = 0.18(2σ -1) 1-σ . In 2018, Bondarenko and Seip [START_REF] Bondarenko | Note on the resonance method for the Riemann zeta function, 50 years with Hardy spaces[END_REF] showed that ν(σ) 1/(2 -2σ) and have the asymptotic behavior

ν(σ) = ( √ 2 2 + o(1)) | log(σ -1 2 )| σ 1 2 , (1 -σ) -1 + O(| log(1 -σ)|) σ 1, (0.3.6) for σ ∈ ( 1 2 + 1 log 2 T , 1 -1 log 2 T ). Here σ 1 2 means σ tends to 1 2 from above with 1 2 + 1 log 2 T σ 3 4
and σ 1 means σ tends to 1 from below with 3 4 σ 1 -1 log 2 T , as T → ∞. By applying the method of de la Bretèche and Tenenbaum [START_REF] De La Bretèche | Sommes de Gál et applications[END_REF], we are able to improve the constant

√ 2 2 to √ 2 in the first assertion of (0.3.6). Theorem 0.3.2. (Corollary 4.1.4) Let 0 < β < 1. For T → ∞ and σ > 1 2 + 1 log 2 T , we have max t∈[T β ,T ] log |ζ(σ + it)| ν β (σ) (log T ) 1-σ (log 2 T ) σ
holds for a function c(σ) which has the asymptotic behavior

ν β (σ) = ( √ 2 + o(1))(1 -β) 1-σ | log(σ -1 2 )|, as σ 1 2 .
Thus, when β 0, we can choose

ν(σ) = ( √ 2 + o(1)) | log(σ -1 2 )|.

Outline of this thesis

Chapter 0 is an introduction in which we present our main results of this thesis.

In Chapter 1 we introduce some preliminary knowledge, including some fundamental properties of the Riemann zeta function we need to use in the following chapters.

Chapters 2-5 are the main parts of this thesis:

In Chapter 2, we study the distribution of large values of |ζ(1 + it)|.

In Chapter 3, we study the distribution of large values of |ζ(σ +it)| in the strip 1 2 < σ < 1. In Chapter 4 (joint with Bin Wei), we study large values of the Riemann zeta function on the 1-line and in the strip 1 2 < e s < 1 separately. In Chapter 5 (joint with Bin Wei), we study large values of the derivatives of the Riemann zeta function on the 1-line.

Preliminaries 1.Basic properties of the Riemann zeta function

For e s > 1, the Riemann zeta function is defined to be the Dirichlet series with all coefficients equal to 1:

ζ(s) = n 1 1 n s .
It has an analytical continuation to the whole complex plane C except s = 1. For s = 1, it has a simple pole with residue 1. Also for e s > 1, it can be written as the Euler product

ζ(s) = p 1 - 1 p s -1
.

For other values of s, the Riemann zeta function can not be written as the Dirichlet series or the Euler product. However, it can be approximated by the truncated forms, says the Dirichlet polynomials and the short Euler products. More precisely, for a suitable x, we could have

ζ(s) ∼ n x 1 n s ,
and for a suitable y, we could have

ζ(s) ∼ ζ(s; y) := p y 1 - 1 p s -1
.

For the first kind of approximation, we have the following asymptotic formula.

Proposition 1.1.1. Let x > 0 be a large number and s = σ + it. Then uniformly in the range σ σ 0 > 0 and |t| x, we have

ζ(s) = n x 1 n s - x 1-s 1 -s + O 1 x σ .

Zero-density estimates

Let 0 σ 1 and T > 0. Define

N (σ, T ) = #{ρ = β + it : β σ, |t| T, ζ(ρ) = 0}.
We have the following estimate

N (σ, T ) T A(σ)(1-σ)+ε ,
where the constant depends only on ε. The Riemann-von Mangoldt formula implies that

N ( 1 2 , T ) T 1+ε , while for 1 2 < σ < 1 trivially we have A(σ)(1 -σ) 1. The zero-density hypothesis states that A(σ) 2 ( 1 2 σ 1),
which is a direct result of RH, but is as powerful as RH in using.

The best-known upper bound of A(σ) uniformly for

A(σ) 3 3σ-1 .
Both estimates suggest that the best-known constant upper bound of A(σ) is 12 5 which is attended at σ = 3 4 . Many improvements have been established for σ near 1. For example, in 2000 Bourgain [START_REF] Bourgain | On large values estimates for Dirichlet polynomials and the density hypothesis for the Riemann zeta function[END_REF] showed that the zero-density hypothesis holds for σ > 25 32 .

Resonance method

The resonance method was introduced by Hilberdink and Soundararajan independently to detect large values of the Riemann zeta function. The principle is to find a Dirichlet polynomial R(t) = n N r(n)n it resonating with ζ(σ + it). Then by the inequality max

T β t T |ζ(σ + it)| | T T β ζ(σ + it)|R(t)| 2 w(t) dt| T T β |R(t)| 2 w(t) dt , (1.3.1)
it remains to choose suitable coefficients r(n) to make the numerator large while the denominator small. The weight function w(t) is to make the dominant contribution arise from the integral in the interval [T β , T ]. Thus we can extend the integral to the whole real numbers without changing the order of magnitude. Moreover, the weight function w(t) is often chosen to be the adjusted Gaussian

w(t) = φ( t log T T ) = exp(-t 2 (log T ) 2 2T 2
), since the Gaussian φ(t) = e -t 2 /2 has a very nice Fourier transform

φ(ξ) = R φ(t)e -itξ dt = √ 2πφ(ξ).
For any n ∈ N + , we have

R 1 n it φ(t) dt = R φ(t)e -it log n dt = φ(log n) > 0.
So if each r(n) is chosen to be positive, then after extending the integral of the numerator in (1.3.1) to R, each term in the expansion will be positive.

GCD sums 1.4.1 Definition

Let M denote a set of positive integers. The GCD sum associated to M is defined as

S σ (M) := m,n∈M gcd(m, n) σ lcm(m, n) σ = m,n∈M gcd(m, n) 2σ (mn) σ ,
and the topic around it is to investigate the upper bound of the quantity

Γ σ (N ) := sup |M|=N S σ (M) N . (1.4.1)
When σ = 1, this was a prize problem by the Dutch Mathematical Society in 1947 suggested by Erdös, and was solved by Gál [START_REF] Gál | A theorem concerning Diophantine approximations[END_REF] in 1949. He showed that

Γ 1 (N ) (log 2 N ) 2 ,
which makes the GCD sum also known as "Gál-type sums".

In 2017, Lewko and Radziw l l [START_REF] Lewko | Refinements of Gál's theorem and applications[END_REF] used the method of probabilistic models to give a much easier proof of Gál's theorem as well as determine the implied constant. They proved that

Γ 1 (N ) = 1 ζ(2) + o(1) e 2γ (log 2 N ) 2 ,
as N → ∞. If we write c = (c 1 , . . . , c N ) ∈ C N , and its norm ||c|| 2 = N j=1 |c j | 2 , then the spectral norm of the GCD matrix (gcd(m, n) σ /lcm(m, n) σ ) is defined by

Q σ (M) := sup c∈C |M| ||c||=1 m,n∈M c m c n gcd(m, n) σ lcm(m, n) σ .
In 2015, Aistleitner, Bondarenko and Seip [START_REF] Aistleitner | GCD sums from Poisson integrals and systems of dilated functions[END_REF] showed that

Γ 1 2 (N ) sup |M|=N Q 1 2 (M) (e 2 + 1)(log N + 2) max n N Γ 1 2 (n).
In this chapter, we focus only on the lower bounds of Γ σ (N ) for 1 2 < σ < 1 for later use. We refer to the paper [START_REF] De La Bretèche | Sommes de Gál et applications[END_REF] of de la Bretèche and Tenenbaum for the completely solution of

Γ 1 2 (N ): as N → ∞, Γ1 2 (N ) = exp (2 √ 2 + o(1)) log N log 3 N log N .

Special case of square-free numbers

The following proposition is a simple construction of M consisting of only square-free integers which was firstly used by Gál [START_REF] Gál | A theorem concerning Diophantine approximations[END_REF] to give lower bounds of GCD sums.

Proposition 1.4.1. Let 1 2 σ 1, N = 2 r
, and M = {p δ 1 1 . . . p δr r : δ j = 0, 1(1 j r)}, the square-free numbers with only the first r primes as prime factors, then we have

S σ (M) = N 1 j r 1 + 1 p σ j .
Proof. Firstly, we fix m ∈ M. Without loss of generality, we can assume m = p 1 p 2 . . . p l with 1 l r. Then we have

n∈M gcd(m, n) σ lcm(m, n) σ = δ 1 ,...,δr∈{0,1} gcd(p 1 p 2 . . . p l , p δ 1 1 . . . p δr r ) σ lcm(p 1 p 2 . . . p l , p δ 1 1 . . . p δr r ) σ = δ 1 ,...,δr∈{0,1} p δ 1 1 . . . p δ l l p 1 . . . p l p δ l+1 l+1 . . . p δr r σ = 1 (p 1 . . . p l ) σ δ 1 ,...,δ l ∈{0,1} (p δ 1 1 . . . p δ l l ) σ δ l+1 ,...,δr∈{0,1} 1
(p δ l+1 l+1 . . . p δr r ) σ = 1 j l 1 p σ j 0 j l (1 + p σ ) l+1 j r 1 + 1 p σ j = 1 j r 1 + 1 p σ j ,
which does not depend on m. We have

S σ (M) = m∈M n∈M gcd(m, n) σ lcm(m, n) σ = 1 j r 1 + 1 p σ j m∈M 1 = N 1 j r 1 + 1 p σ j , since |M| = 2 r = N , which completes the proof.
With the help of the above proposition, when 1 2 < σ < 1, we can simply get a nontrivial lower bound Γ σ (N )

1 j log N/ log 2 1 + 1 p σ j = exp 1 j log N/ log 2 log 1 + 1 p σ j exp 1 j log N/ log 2 1 p σ j + C exp 1 (1 -σ)(log 2) 1-σ + o(1) (log N ) 1-σ (log 2 N ) σ , (1.4.2)
by the definition of Γ σ (N ) and Lemma 4.2.2.

Gál's identity

Let p 1 , . . . , p r be r different primes. Define

M = M(r, l) := {m = p v 1 1 . . . p vr r : 0 v j l -1 (1 j r)}.
Gál [START_REF] Gál | A theorem concerning Diophantine approximations[END_REF] showed the following identity, which can be seen as a generalization of the case of square-free numbers:

S σ (M) = 1 j r l + 2 1 v l-1 l -v p vσ j = |M| 1 j r 1 + 2 1 v l-1 1 - v l 1 p vσ j . (1.4.3)
We have 1 + 2

1 v l-1 1 - v l 1 p vσ j = 2 v 0 - v l 1 - v l 1 p vσ j -1 = 2 1 - 1 p σ j -1 - 2 lp σ j 1 - 1 p σ j -1 -1 + O(p -lσ j ) = 1 - 1 p σ j -1 1 + 1 p σ j - 2 lp σ j + O(p -lσ j ).
Thus (1.4.3) turns to be

S σ (M) |M| = 1 j r 1 - 1 p σ j -1 1 + 1 p σ j - 2 lp σ j + O(p -lσ j ) = 1 j r 1 - 1 p σ j - 2
1 j r 1 - 1 p σ j 1 + 1 p σ j - 2 lp σ j + O(p -lσ j ) = 1 j r 1 - 1 p σ j -2 1 j r 1 - 1 p 2σ j + O 1 lp σ j + O(p -lσ j ) . (1.4.4) Since 1 j r 1 - 1 p 2σ j = 1 ζ(2) + o(p 1-2σ r ),
The second product of (1.4.4) can be written as

1 j r 1 - 1 p 2σ j + O 1 lp σ j + O(p -lσ j ) = 1 ζ(2) + o(p 1-2σ r ) + O 1 l 1 j r 1 p σ j .
We choose l = r + log N and r such that

(r + log N ) r N < (r + 1 + log N ) r+1 .
Taking the logarithm of the above inequality, we have

r log(r + log N ) log N < (r + 1) log(r + 1 + log N ).
It follows that

r = (1 + o(1)) log N log 2 N . When σ = 1, (1.4.4) gives the best bound of Lewko-Radziwi l l [35] Γ 1 (N ) = max |M|=N S σ (M) |M| 1 ζ(2) + o(1) e 2γ (log 2 N ) 2 .
However, when 1 2 < σ < 1, the choice of M can not provide a good lower bound for Γ σ (N ).

Special case of friable numbers

If we restrict M to be a set of friable numbers, then we can easily give an upper bound for the GCD sums attached to M.

Proposition 1.4.2. For 0 < σ < 1, assume M ⊂ S(y), where S(y) denotes the set of y-friable numbers. Then we have

S σ (M) |M| p y 1 + 2 p σ -1 .
Proof. Let p 1 , p 2 , . . . be all the primes in ascending order. Assume p k = max m∈M P + (m).

For any fixed m ∈ M, assume m = k j=1 p u j j with u j 0 for each 1 j k. We have

n∈M gcd(m, n) lcm(m, n) σ v 1 0 • • • v k 0 gcd(p u 1 1 . . . p u k k , p v 1 1 . . . p v k k ) lcm(p u 1 1 . . . p u k k , p v 1 1 . . . p v k k ) σ = v 1 0 • • • v k 0 k j=1 gcd(p u j j , p v j j ) lcm(p u j j , p v j j ) σ = k j=1 v j 0 gcd(p u j j , p v j j ) lcm(p u j j , p v j j ) σ = k j=1 v j 0 p min(u j ,v j ) j p max(u j ,v j ) j σ = k j=1 v j 0 p -σ|u j -v j | j k j=1 v j 0 p -σ|u j -v j | j k j=1 1 + 2 v 1 p -σv j = k j=1 1 + 2 p σ j -1 .
Since the last product does not depend on m, we have

S σ (M) = m∈M n∈M gcd(m, n) lcm(m, n) σ m∈M k j=1 1 + 2 p σ j -1 = k j=1 1 + 2 p σ j -1 m∈M 1 =|M| k j=1 1 + 2 p σ j -1 |M| p y 1 + 2 p σ -1 ,
which completes the proof.

Probabilistic models 1.5.1 Models for the Riemann zeta function

For each prime p, let X(p) be the random variable uniformly distributed on the unit circle. And they are independently identically distributed for all distinct primes. Define the random Euler product

ζ(σ, X) := p 1 - X(p) p σ -1
, which is almost surely convergent for σ > 1 2 . Extending the definition of X(n) for any integer n by

X(n) := p v ||n X(p) v ,
then we have the orthogonality

E(X(m)X(n)) = 1 if m = n, 0 otherwise.
And for σ > 1 2 we can also write ζ(σ, X) as

ζ(σ, X) = n 1 X(n) n σ .
It turns out that ζ(σ, X) is a good model for both the Riemann zeta function ζ(σ + it) and the Dirichlet L-function.

Proposition 1.5.1. Denote the short random Euler product by ζ(σ, X; y) = p y (1 -X(p)/p σ ) -1 . Then for y (log T ) 2 , and k log T /y 1-σ , we have

1 T 2T T |ζ(σ + it; y)| k dt = E(|ζ(σ, X; y)| k ) + O exp - log T 4 log y .
For y (log q) 2 , and k log q/y 1-σ , we have

1 ϕ(q) χ(mod q) |L(σ, χ; y)| k = E(|ζ(σ, X; y)| k ) + O exp - log q 4 log y .

Models for the GCD sum

Using the definitions of X and ζ(σ, X) as in the last section, for any finite set of positive integers M, we define D(X, M) := m∈M X(m).

We will see that ζ(σ, X) 2 D(X, M) 2 is a good model for S σ (M).

Proposition 1.5.2. We have

E(|ζ(σ, X)D(X, M)| 2 ) = ζ(2σ)S σ (M).
Proof. Expanding the square, we have

|ζ(σ, X)D(X, M)| 2 = l,k 1 X( )X(k) (lk) σ m,n∈M X(m)X(n) = l,k 1 m,n∈M X(lm)X(kn) (lk) σ .
So by the orthogonality of X, we have

E(|ζ(σ, X)D(X, M)| 2 ) = E l,k 1 m,n∈M X(lm)X(kn) (lk) σ = l,k 1 m,n∈M E(X(lm)X(kn)) (lk) σ = l,k 1 m,n∈M lm=kn 1 (lk) σ = m,n∈M l,k 1 lm=kn 1 (lk) σ .
For any fixed m, n, the relation lm = kn implies

l m gcd(m, n) = k n gcd(m, n) , while gcd m gcd(m, n) , n gcd(m, n) = 1.
Thus we must have

l = j n gcd(m, n) and k = j m gcd(m, n) ,
for some positive integer j. Then we have

m,n∈M l,k 1 lm=kn 1 (lk) σ = m,n∈M j 1 gcd(m, n) 2 j 2 mn σ = j 1 1 j 2σ m,n∈M gcd(m, n) 2σ (mn) σ = ζ(2σ)S σ (M).
This completes the proof.

Models for L(1, χ d )

Since the real primitive character χ d only takes values ±1 and 0, the model X in the last two sections is not in accordance with χ d . We define a new random variable X(p) for each prime p:

X(n) = ±1 Prob = p 2(p+1) , 0 Prob = 1 p+1 .
And again we extend the definition to any integers completely multiplicatively

X(n) = p v ||n X(p) v .
Then we similarly define

L(σ, X) = p 1 - X(p) p σ -1
, and it will be a good model for L(σ, χ d ).

Proposition 1.5.3. Denote the short random Euler product by L(σ, X; y) = p y (1 -X(p)/p σ ) -1 . Then for y (log x) 2 , and k log x log 2 y/y 1-σ , we have the following asymptotic formula

1 x/ζ(2) |d| x L(σ + it, χ d ; y) k dt = E(L(σ, X; y) k ) + O exp - log x log 2 y 40 log y . 2 Distribution of |ζ(1 + it)| 2.

Background

The study of the value distribution of the Riemann zeta function ζ(s) can date back to the early twentieth century when Bohr showed that for any z ∈ C * and ε > 0, there are infinitely many s's with 1 < e s < 1 + ε such that ζ(s) = z. Later in 1932, Bohr and Jessen [START_REF] Bohr | Über die Werteverteilung der Riemannnschen Zetafunktion[END_REF] showed that log ζ(σ + it) has a continuous distribution on the complex plane for any σ > 1 2 . The values on the 1-line have much significance. For example, the fact that ζ(1 + it) = 0 implies the prime number theorem. The extreme values of ζ(1 + it) has been widely investigated. In 1925, Littlewood [START_REF] Littlewood | On the Riemann zeta-function[END_REF] showed that there exists arbitrarily large t for which

|ζ(1 + it)| {1 + o(1)}e γ log 2 t.
Here and throughout, we denote by γ the Euler constant and by log j the j-th iterated logarithm. In 1972, Levinson [34] improved the error term from o(log 2 t) to O(1), by showing that there exists arbitrarily large t such that

|ζ(1 + it)| e γ log 2 t + O(1).
In 2006, Granville and Soundararajan [START_REF] Granville | The Riemann Zeta Functionn and Related Themes: Papers in Honour of Professor K. Ramachandra[END_REF] This bound is best possible up to the error term O(1), since in [START_REF] Granville | The Riemann Zeta Functionn and Related Themes: Papers in Honour of Professor K. Ramachandra[END_REF], Granville and Soundararajan conjectured that max

t∈[T,2T ] |ζ(1 + it)| = e γ {log 2 T + log 3 T + C 0 + 1 -log 2 + o(1)}, (2.1.2)
where C 0 is some absolute constant (see (2.1.5) below). This conjecture was based on some analysis on the following distribution function they introduced in [START_REF] Granville | The Riemann Zeta Functionn and Related Themes: Papers in Honour of Professor K. Ramachandra[END_REF]: define for T > 1,

Φ T (τ ) := 1 T meas t ∈ [T, 2T ] : |ζ(1 + it)| > e γ τ . (2.1.3)
Then they proved the asymptotic formula in the logarithm of the distribution function

Φ T (τ ) = exp - 2e τ -C 0 -1 τ 1 + O 1 √ τ + e τ log T (2.1.4)
valid uniformly for 1 τ < log 2 T -20, where

C j := 2 0 log t 2 j log I 0 (t) t 2 dt + ∞ 2 log t 2 j log I 0 (t) -t t 2 dt (2.1.5)
and

I 0 (t) := ∞ n=0 (t/2) 2n (n!) 2 • (2.1.6)
The distribution function (2.1.4) describes the frequency with which each large value is attained. Obviously, the maximum of the range of τ is much less than the large value (2.1.1). However, if (2.1.4) were to persist to the end of the viable range, then we could get (2.1.2). The method to prove (2.1.4) was also adjusted to apply to the distribution of values on the 1-line of other L-functions. Also in [START_REF] Granville | The Riemann Zeta Functionn and Related Themes: Papers in Honour of Professor K. Ramachandra[END_REF], Granville and Soundararajan showed that the distribution of the Dirichlet L-functions in the aspect of the modulo of characters has the same form as (2.1.4). This result can be used to study the distribution of large character sums, see [START_REF] Bober | The distribution of the maximum of character sums[END_REF] and [START_REF] Bober | The frequency and the structure of large character sums[END_REF]. In 2003, Granville and Soundararajana [START_REF] Granville | The distribution of values of L(1, χ d )[END_REF] studied the distribution of the Dirichlet L-functions of quadratic characters L(1, χ d ), which proves part of Montgomery and Vaughan's conjecture in [START_REF] Montgomery | Extreme values of the Riemann zeta function[END_REF]. In 2007, Wu [START_REF] Wu | Note on a paper by A[END_REF] improved this result by giving a high order expansion in the exponent of the distribution function. In 2008, Liu, Royer and Wu [START_REF] Liu | On a conjecture of Montgomery-Vaughan on extreme values of automorphic L-functions at 1, Anatomy of integers[END_REF] considered the case of symmetric power L-functions. In 2010, Lamzouri [START_REF] Lamzouri | Distribution of values of L-functions at the edge of the critical strip[END_REF] studied a generalized L-function which can cover the results of [START_REF] Granville | The distribution of values of L(1, χ d )[END_REF][START_REF] Liu | On a conjecture of Montgomery-Vaughan on extreme values of automorphic L-functions at 1, Anatomy of integers[END_REF]. Again concerning the Riemann zeta function, in 2008 Lamzouri [START_REF] Lamzouri | The two dimensional distribution of values of ζ(1 + it)[END_REF] Inspired by the result of Wu [START_REF] Wu | Note on a paper by A[END_REF], the aim of this chapter is to get an improvement of (2.1.4), which presents a higher order expansion in the exponent.

Theorem 2.1.1. There is a sequence of real numbers {a j } j 1 such that for any integer J 1 we have

Φ T (τ ) = exp - 2e τ -C 0 -1 τ 1 + J j=1 a j τ j + O J 1 τ J+1 + e τ log T uniformly for T → ∞ and 1 τ log 2 T -20, where C 0 is defined as in (2.1.5). Moreover, a 1 = 2(1 + C 0 -C 1 ).
Our main new ingredient for the proof of Theorem 2.1.1 is Proposition 2.5.1 below, which gives a better approximation of the distribution function of the short Euler products:

Φ T (τ ; y) := 1 T meas t ∈ [T, 2T ] : |ζ(1 + it; y)| > e γ τ , (2.1.7) where ζ(s; y) := p y (1 -p -s ) -1 . (2.1.8)
For this, it is necessary to improve Theorem 3 of [START_REF] Granville | The Riemann Zeta Functionn and Related Themes: Papers in Honour of Professor K. Ramachandra[END_REF] (see Propositions 2.3.1 and 2.4.1 below).

Preliminary lemmas

Let k 1 be a positive integer. Define d k (n) by the relation

ζ(s) k = n 1 d k (n)n -s ( e s > 1). (2.2.1)
Firstly, we quote the following asymptotic formulae of sums attached to the divisor function d k (n) and the Bessel function I 0 (t) to show their correlation.

Lemma 2.2.1. For any prime p and positive integer k, we have

ν 0 d k (p ν ) 2 p 2ν = I 0 2k p exp O k p 2 , (2.2.2) min(1, p/k) 50 1 - 1 p -2k ν 0 d k (p ν ) 2 p 2ν 1 - 1 p -2k , (2.2.3)
where I 0 (t) is the Bessel function as defined in (2.1.6).

Proof. This is [26, Lemma 4].

We need to approximate Riemann zeta function by its short Euler product. The following lemma shows that when ζ(s) has no zero in a good region, it can be approximated well by its short Euler product. Further for σ 0 < σ 1, we have

log ζ(σ + it) = y n=2 Λ(n) n σ+it log n + O log |t| (σ 1 -σ 0 ) 2 y σ 1 -σ , where σ 1 = min σ 0 + 1 log y , σ+σ 0 2 . Proof. See [26, Lemma 1].
In order to approximate the Riemann zeta function ζ(s) by its truncated Euler product ζ(s; y) defined by (2.1.8), we need the following evaluation for moments of the sum over complex power of primes between two large numbers. By the choice of the value of k, we have

1 T 2T T y p z 1 p 1+it 2k dt log T 150y log y log 2 T k + T -1/3 √ log T 12 y log y log 2 T 2k + T -1/3 √ log T 12 √ ylog 2 T 2k + T -1/3 ,
where the last inequality holds since y log T . Thus

y p z 1 p 1+it √ log T √ y log 2 T
for all t ∈ [T, 2T ] except for a set of measure 12 -2k T = T exp(-log 12 log T /(150 log 2 T ))

T exp(-log T /(49 log 2 T )). Combining this with the first step, Lemma 2.2.4 follows.

An asymptotic development

The integer n 1 is called y-friable if the largest prime factor P (n) of n is less than y (P (1) = 1 by convention). Denote by S(y) the set of y-friable integers and define

D k (y) := n∈S(y) d k (n) 2 /n 2 .
(2.3.1)

The aim of this section is to prove the following proposition, which is an improvement of the second part of Theorem 3 in [START_REF] Granville | The Riemann Zeta Functionn and Related Themes: Papers in Honour of Professor K. Ramachandra[END_REF]. Our improvement is double: a higher order expansion in the exponent and a larger domain of y.

Proposition 2.3.1. Let A > 1 be a positive number and let J 0 be an integer. We have

D k (y) = p k 1 - 1 p -2k exp 2k log k J j=0 C j (log k) j + O A,J 1 (log k) J+1 + k y
uniformly for k 2 and 2k y k A , where the C j is defined as in (2.1.5) and the constant implied depends on A and J only.

Proof. Firstly we have trivially

√ k<p y exp O k p 2 = e O( √ k) , p √ k min(1, p/k) 50 = e O( √ k) . (2.3.2)
Secondly, by Lemma 2.2.1, we can write

D k (y) = p √ k 1 - 1 p -2k √ k<p y I 0 2k p e O( √ k) = p k 1 - 1 p -2k Π 1 Π 2 e O( √ k) , (2.3.3)
where

Π 1 := √ k<p k 1 - 1 p 2k I 0 2k p and Π 2 := k<p y I 0 2k p .
In order to evaluate Π 1 , we apply the formula log(1

+ t) = t + O(t 2 ) (|t| 1
2 ) and the first estimate in (2.3.2) to obtain

Π 1 = exp √ k<p k log I 0 2k p - 2k p + O √ k .
Recall the prime number theorem

π(u) := p u 1 = u 2 du log u + O(ue -2c √ log u ).
Then we can derive that

√ k<p k log I 0 2k p - 2k p = k √ k log I 0 2k u - 2k u dπ(u) = k √ k log I 0 (2k/u) -2k/u log u du + O ke -c √ log k .
(2.3.4)

By putting t = 2k/u and using the fact that

1 1 -t = J j=0 t j + O J (t J+1 ) (|t| 1 2 ),
we can derive that the integral in (2.3.4) is equal to

2k log k 2 √ k 2 log I 0 (t) -t t 2 (1 -log(t/2) log k ) dt = 2k log k J j=0 C * j (k) (log k) j + O J C * J+1 (k) (log k) J+1 , (2.3.5)
where

C * j (k) := 2 √ k 2 log t 2 j log I 0 (t) -t t 2 dt.
Since log I 0 (t) = t + O(log t) for t 2, we have

∞ 2 √ k log t 2 j log I 0 (t) -t t 2 dt ∞ √ k (log t) j+1 t 2 dt j (log k) j+1 √ k ,
which implies that

C * j (k) = C * j + O j (log k) j+1 √ k with C * j := ∞ 2 log t 2 j log I 0 (t) -t t 2 dt.
Combining this with (2.3.4) and (2.3.5), we obtain

√ k<p k log I 0 2k p - 2k p = 2k log k J j=0 C * j (log k) j + O J 1 (log k) J+1
.

Therefore we derive that

Π 1 = exp 2k log k J j=0 C * j (log k) j + O J 1 (log k) J+1 . (2.3.6)
For Π 2 , by the prime number theorem, we have similarly

k<p y log I 0 2k p = y k log I 0 2k u du log u + O ke -c √ log k = 2k log k J j=0 C * * j (log k) j + O k y + O ke -c √ log k ,
where

C * * j := 2 0 log t 2 j log I 0 (t) t 2 dt.
Here we have used the inequality log I 0 (t) t 2 for 0 t < 2 to evaluate the truncated integral that 2k/y 0 log t 2

j log I 0 (t) t 2 dt k/y 0 (log t) j dt A,j k(log k) j y •
Therefore we derive that Proposition 2.4.1. Let A > 0 be a constant. Then we have

Π 2 = exp 2k log k J j=0 C * * j (log k) j + O A,J k y . ( 2
1 T 2T T |ζ(1 + it; y)| 2k dt = D k (y) 1 + O A exp - log T 2(log 2 T ) 4 uniformly for      T T 0 (A), e 2 log T y (log T )(log 2 T ) A , k ∈ N ∩ [2, (log T )/(e 10 log(y/ log T ))], (2.4.1)
where the implied constant and the constant T 0 (A) depend on A only.

We show that for k and y in (2.4.1), the diagonal terms lead to the main term, while the off-diagonal terms only contribute to the error term. For this, we need to establish a preliminary lemma.

A preliminary lemma

If 2 k 10 6 , we write I 0 = (k, y], I 1 = (1, k] and J = 0. When k > 10 6 , we take J = 4(log 2 k)/ log 2 and divide (1, y] into J + 2 intervals

(1, y] = I 0 ∪ I 1 ∪ • • • ∪ I J+1 ,
where I 0 := (k, y], I j := (k/2 j , k/2 j-1 ] (1 j J) and

I J+1 := (1, k/2 J ] ⊂ (1, 2k/(log 2 k) 4 ].
For each j ∈ {0, 1, . . . , J + 1}, we use S(I j ) to represent the set of all positive integers which have prime divisors only in I j (1 ∈ S(I j ) by convention). Recall that S(y) is the set of y-friable integers. Thus

n ∈ S(y) ⇔ n uniquely = n 0 • • • n J+1 with n j ∈ S(I j ) (0 j J + 1). (2.4.2) Set D k,j := h∈S(I j ) d k (h) 2 /h 2 (2.4.3) such that D k (y) = D k,0 D k,1 • • • D k,J+1 . (2.4.4)
We have the following lemma.

Lemma 2.4.2. Let G 0 := T 1/5 and G j := T 1/(5j 2 ) (j 1). Then we have

g∈S(I j ), g>G j 2 ω(g) g h∈S(I j ) d k (gh)d k (h) h 2 D k,j exp - log T (log 2 T ) 4
(2.4.5)

for (T, y, k) in (2.4.1) and 0 j J + 1, where ω(n) denotes the number of distinct prime factors of n.

Proof. This is essentially [26, Lemma 5]. The difference is that the upper bound of y is shifted from (log T )(log So this makes no difference so that we can follow Granville and Soundararajan's procedure.

Here we reproduce their proof for convenience of the reader. Denote by S k (I j ) the member on the left-hand side of (2.4.5).

Firstly we consider the case of 1 j J + 1. For δ = 1/(2 j/2 log k), by Rankin's trick and exchanging the order of the sums, we have

S k (I j ) 1 G δ j g∈S(I j ) 2 ω(g) g 1-δ h∈S(I j ) d k (gh)d k (h) h 2 = 1 G δ j h∈S(I j ) d k (h) h 1+δ g∈S(I j ) 2 ω(g) d k (gh) (gh) 1-δ • (2.4.7)
The inner sum is over part of S(I j ), so for any h ∈ S(I j ) we have

g∈S(I j ) 2 ω(g) d k (gh) (gh) 1-δ g∈S(I j ) 2 ω(gh) d k (gh) (gh) 1-δ g∈S(I j ) 2 ω(g) d k (g) g 1-δ • Thus S k (I j ) 1 G δ j h∈S(I j ) d k (h) h 1+δ g∈S(I j ) 2 ω(g) d k (g) g 1-δ = 1 G δ j p∈I j 1 - 1 p 1+δ -k 2 1 - 1 p 1-δ -k -1 1 G δ j p∈I j 2 1 - 1 p -2k Ξ j (p) -k (2.4.8) with Ξ j (p) := 1 - 1 p 1+δ 1 - 1 p 1-δ 1 - 1 p -2 = 1 - p(p δ + p -δ -2) (p -1) 2 •
Noticing that p ∈ I j with 1 j J + 1, we have p k. Thus by the first inequality in (2.2.3) of Lemma 2.2.1, we can derive that

p∈I j 1 - 1 p -2k D k,j p∈I j 50k p , (2.4.9)
while We choose δ = 1/(2 j/2 log k). For 1 j J + 1, we have

Ξ j (p) = 1 - 2p (p -1) 2 ∞ n=1 (δ log p) 2n (2n)! 1 - 2p(δ log p) 2 (p -1) 2 ∞ n=1 2 -j(n-1) (2n)! 1 - c(2δ log p) 2 p • (2.4.10) with c := ∞ n=1 2 -(n-1) (
p∈I j log 100k p + √ 3k(2δ log p) 2 p log 100k k/2 j + √ 3k(2δ log(k/2 j )) 2 k/2 j 2(k/2 j-1 ) log(k/2 j-1 ) (j log 2 + 2 log 10 + 4 √ 3) 4k 2 j log k 50jk 2 j log k 25j log T 2 j e 10 log k
25 log T 2 j/2 j 2 e 6 log k thanks to (2.4.6) and the inequality j 3 e 4 2 j/2 (j 1). Thus

log 1 G δ j + p∈I j log 100k p + √ 3k(2δ log p) 2 p - 1 -125e -6 5j 2 2 j/2 • log T log k - 1 -125e -6 5J 2 2 J/2 • log T log k - log T (log k) 4 - log T (log 2 T ) 4
(2.4.12) for (T, y, k) in (2.4.1) and 1 j J + 1. Inserting (2.4.12) into (2.4.11), then 1 j J + 1 we have

S k (I j ) D k,j exp - log T (log 2 T ) 4 .
Thus the lemma in this case follows.

When j = 0, by Rankin's trick and the trivial inequality d k (gh) d k (g)d k (h), we have for any 0 < δ < 1

S k (I 0 ) D k,0 G δ 0 g∈S(I 0 ) 2 ω(g) d k (g) g 1-δ = D k,0 G δ 0 p∈I 0 2 1 - 1 p 1-δ -k -1 .
(2.4.13)

For k < p y, we have the upper bound

2 1 - 1 p 1-δ -k -1 2 exp 2k p 1-δ -1 exp 3k p 1-δ exp 3ky δ log k • log p p .
Inserting this into (2.4.13) and using [START_REF] Tenenbaum | Introduction to analytic and probabilistic number theory[END_REF]Theorem I.1.7]

S k (I 0 ) D k,0 exp -δ log G 0 + 20e 1/5 log T e 10 log 2 T
for (T, y, k) in (2.4.1). Now the result of Lemma 2.4.2 follows by recalling that G 0 = T 1/5 .

Proof of Proposition 2.4.1

As in [START_REF] Granville | The Riemann Zeta Functionn and Related Themes: Papers in Honour of Professor K. Ramachandra[END_REF], we shall prove a more general result: Let A > 0 be a constant and R ⊂ {0, 1, . . . , J + 1}. Then we have

1 T 2T T |ζ(1 + it; R)| 2k dt = D k (R) 1 + O A exp - log T 2(log 2 T ) 4
(2.4.14)

uniformly for (T, y, k) in (2.4.1), where

I R := r∈R I r , ζ(s; R) := p∈I R (1 -p -s ) -1 , D k (R) := n∈S(I R ) d k (n) 2 /n 2 (2.4.15)
and the implied constant depends on A only. We shall prove (2.4.14) by induction on the cardinal of R. The case of R = ∅ (i.e. |R| = 0) is trivial, since ζ(s; ∅) = 1 = D k (∅). Now we suppose that (2.4.14) holds for all proper subset of R and prove that it is true for R.

Firstly, in view of (2.4.2), we have

1 T 2T T |ζ(1 + it; R)| 2k dt = m,n∈S(I R ) d k (m)d k (n) mn 1 T 2T T n m it dt = mr,nr∈S(Ir) r∈R r∈R d k (m r )d k (n r ) m r n r 1 T 2T T r∈R n r m r it dt.
(2.4.16) Denote g j = lcm(m j , n j ) gcd(m j , n j ) and h j = gcd(m j , n j ).

Using the principle of inclusion-exclusion, we divide the sum in (2.4.16) where the G j is defined as in Lemma 2.4.2.

In the first sum, the case g r = 1 (r ∈ R) counts the diagonal terms and leads to the main term

n∈S(I R ) d k (n) 2 /n 2 = D k (R).
Otherwise, we have r∈R (m r /n r ) = 1. Therefore by g r G r we have

log r∈R m r n r = log r∈R m r /h r n r /h r log 1 + r∈R g -1 r r∈R G -1 r .
Thus in these terms we have

1 T 2T T r∈R m r n r it dt 1 T log r∈R m r n r -1 1 T 2/5 •
Therefore the sum over these terms is

1 T 2/5 m,n∈S(I R ) d k (m)d k (n) mn 1 T 2/5 p∈I R 1 - 1 p -2k
.

By (2.2.3) in Lemma 2.2.1 and the inequality -log(1 -t) 2t (0 t 1 2 ), we have

p∈I R 1 - 1 p -2k p∈I R p k 50k p ν 0 d k (p ν ) 2 p 2ν p∈I R k<p y 1 - 1 p -2k D k (R) exp p k log 50k p + k<p y 4k p D k (R) exp 10k log k log 25y k .
Therefore, the contribution of the first sum in (2.4.17) is mr,nr∈S(Ir) gr Gr ∀r∈R

= D k (R)+O D k (R) T 2/5 exp 12k log k log 25y k = D k (R) 1+O 1 T 1/
d k (m w )d k (n w ) m w n w • (2.4.20)
For any multiplicative function f , we have f (m)f (n) = f (lcm(m, n))f (gcd(m, n)). While the number of (m, n) such that gcd(m, n)/lcm(m, n) = g, gcd(m, n) = h is 2 ω(g) . Thus we derive that mw,nw∈S(Iw) gw>Gw

d k (m w )d k (n w ) m w n w = mw,nw∈S(Iw) gw>Gw d k (g w h w )d k (h w ) g w h 2 w = mw,nw∈S(Iw) gw>Gw 2 ω(gw) d k (g w h w )d k (h w ) g w h 2 w • (
W⊂R W =∅ D k (W) 1 T 2T T |ζ(1 + it; R W)| 2k dt exp - log T (log 2 T ) 4 .
(2.4.22)

According to the induction hypothesis, it follows that and the definition of its distribution function:

1 T 2T T |ζ(1 + it; R W)| 2k dt D k (R W). (2.4.23) Then noticing that D k (W)D k (R W) = D k (R), (2.4.22) is bounded by 2 L+1 D k (R) exp - log T (log 2 T ) 4 D k (R) exp - log T 2(
Φ T (τ ; y) := 1 T meas t ∈ [T, 2T ] : |ζ(1 + it; y)| > e γ τ .
The aim of this section is to prove the following result.

Proposition 2.5.1. Let A > 0 be any constant and let J 1 be an integer, and ε satisfying (2.5.10). Then we have

Φ T (τ + ε; y) exp - 2e τ -C 0 -1 τ 1 + J j=1 a j τ j + O J 1 τ J+1 + e τ y Φ T (τ -ε; y) (2.5.1)
uniformly for

T T 0 (A), e 2 log T y (log T )(log 2 T ) A , 2 τ log 2 T -20, (2.5.2)
where the a j and C 0 are the same as in Theorem 2.1.1, T 0 (A) is a positive constant depending on A and the implied constant depends on A and J at most.

Two preliminary lemmas

In the following lemma, we will see the correlation between the distribution function and the moments of the short Euler products:

Lemma 2.5.2. Let A > 0 be any constant and let J 1 be an integer. Then we have

∞ 0 Φ T (t; y)t 2κ-1 dt = (log κ) 2κ 2κ exp 2κ log κ J j=0 C j (log κ) j + O A,J κ y + 1 (log κ) J+1
(2.5.3) uniformly for T 2, e 2 log T y (log T )(log 2 T ) A , 2 κ (log T )/(e 10 log(y/ log T )), (2.5.4) where the C j are defined as in (2.1.5) and the implied constant depends on A and J at most.

Proof. For any κ > 0, we have

D ∞ 0 Φ T (t; y)t 2(k-1)-1 dt 2κ-1 2k-3 ∞ 0 Φ T (t; y)t 2κ-1 dt C ∞ 0 Φ T (t; y)t 2k-1 dt 2κ-1 2k-1
.

On the other hand, setting f (u

) := 2u log u J j=0 C j (log u) j , then f (u) = -2 log u J j=0 jC j (log u) j . Thus f (k -1) = f (κ) + O(1) and f (k) = f (κ) + O(1). (2.5.6)
Now we can obtain (2.5.3) for κ / ∈ N by substituting (2.5.3) for integers k -1 and k and by using (2.5.6). This completes the proof of Lemma 2.5.2.

Lemma 2.5.3. Let {a j } j 0 be a sequence of real numbers and J 0 be an integer. If

τ = log κ + a 0 + J j=1 a j (log κ) j + O J 1 (log κ) J+1 (k → ∞), (2.5.7)
then there is a sequence of real numbers {b j } j 0 such that

log κ = τ + b 0 + J j=1 b j τ j + O J 1 τ J+1
(τ → ∞).

(2.5.8)

Further we have b 0 = -a 0 and b 1 = -a 1 .

Proof. We shall reason by recurrence on J. Taking J = 0 in (2.5.7), we have

τ = log κ + a 0 + O 1 log κ (k → ∞).
From this we easily deduce that

log κ = τ -a 0 + O 1 τ (τ → ∞).
This is (2.5.8) with J = 0 and b 0 = -a 0 . Suppose that

τ = log κ + J+1 j=0 a j (log κ) j + O J 1 (log κ) J+3 (k → ∞). (2.5.9)
Clearly this implies (2.5.7). Thus according to the hypothesis of recurrence, (2.5.8) holds. Using (2.5.8) and (2.5.9); we can derive that

log κ = τ -a 0 + J+1 j=1 a j (log κ) j + O J 1 τ J+2 = τ -a 0 + J+1 j=1 a j τ j 1 + J+2-j d=1 b d-1 τ d + O J 1 τ J+2-j -j + O J 1 τ J+2 .
This implies the required result via the Taylor development of (1 -t) -j .

Proof of Proposition 2.5.1

Let ε ∈ [c(log κ) -J-1 , 9c(log κ) -J-1 ] be a parameter to be chosen later, where c is a large constant. Without loss of generality, we can suppose ε (log κ) -J , ε 2 (log κ) -J-1 (2.5.10)

for k k 0 , where κ 0 = κ 0 (c) is a constant depending c. Put K = κe ε . Noticing that ( t τ +ε ) 2K-2κ 1 for t τ + ε, we have 2κ

∞ τ +ε Φ T (t; y)t 2κ-1 dt (τ + ε) 2κ-2K 2K ∞ 0 Φ T (t; y)t 2K-1 dt .
From this and Lemma 2.5.2, we deduce that

∞ τ +ε Φ T (t; y)t 2κ-1 dt ∞ 0 Φ T (t; y)t 2κ-1 dt exp 2(g(K, τ ) -g(κ, τ )) + O J κ 2 y + κ (log κ) J+3
(2.5.11)

uniformly for (T, y, κ) in (2.5.4) above, where

g(κ, τ ) := -κ log τ + ε log κ + κ log κ J+1 j=0 C j (log κ) j •
(2.5.12)

Let τ 0 = τ 0 (c, J) be a suitable constant depending on c and J. For

τ 0 τ log 2 T -20, take κ = κ τ such that τ = log κ + a 0 + J+1 j=1 a j (log κ) j , (2.5.13)
where the a j = a j (C 0 , . . . , C j ) are constants to be determined later. Our choice of τ 0 guarantees that τ τ 0 ⇒ κ κ 0 , which guarantees that ε 9c(log κ) -J-1 (log κ) -J and ε 2 81c 2 (log κ) -2J-2 (log κ) -2J-1 .

(2.5.14)

These bounds will be used often and all implied constants in the O-symbol is independent of c. In view of (2.5.13) and the Taylor formula, we can write

g(κ, τ ) = -κ log 1 + a 0 + ε log κ + 1 log κ J+1 j=1 a j (log κ) j + κ log κ J+1 j=0 C j (log κ) j = -κ a 0 -C 0 + ε log κ + a 1 -a 2 0 -C 1 + a 0 ε (log κ) 2 + J+1 j=2 a j -a * j -C j (log κ) j+1 + O J 1 (log κ) J+3 ,
where the a * j = a * j (a 0 , . . . , a j-1 ) are constants depending on a 0 , . . . , a j-1 . Take

a 0 = C 0 + 1, a 1 = C 2 0 + C 0 + C 1 + 2, a j = a * j (a 0 , . . . , a j-1 ) + C j (2 j J + 1). Thus g(κ, τ ) = -κ 1 + ε log κ + a 1 -a 2 0 -C 1 + a 0 ε (log κ) 2 + O J κ (log κ) J+3 . Let T := log K + a 0 + J+1 j=1 a j (log K) j , then g(K, T) = -K 1 + ε log K + a 1 -a 2 0 -C 1 + a 0 ε (log K) 2 + O J κ (log κ) J+3 .
From these, we easily deduce that

g(K, T) -g(κ, τ ) = -K 1 + ε log K - 1 + ε log κ + a 1 -a 2 0 -C 1 + a 0 ε (log K) 2 - a 1 -a 2 0 -C 1 + a 0 ε (log κ) 2 -(K -κ) 1 + ε log κ + a 1 -a 2 0 -C 1 + a 0 ε (log κ) 2 + O J κ (log κ) J+3 .
Using (2.5.14), a simple computation shows that

g(K, T) -g(κ, τ ) = K (1 + ε)ε (log K) log κ + (a 1 -a 2 0 -C 1 + a 0 ε)ε log(Kκ) (log K) 2 (log κ) 2 -(K -κ) 1 + ε log κ + a 1 -a 2 0 -C 1 + a 0 ε (log κ) 2 + O J κ (log κ) J+3 = e ε εκ (log κ) 2 -(e ε -1)κ 1 + ε log κ + a 1 -a 2 0 -C 1 (log κ) 2 + O J κ (log κ) J+3 .
(2.5.15) On the other hand, in view of (2.5.12), we have

∂g ∂τ (κ, τ ) = - κ τ + ε •
Thus we have, for some η κ ∈ (τ, T),

g(K, τ ) -g(K, T) = ∂g ∂τ (K, η κ )(τ -T) εK τ + ε 1 + O J 1 (log κ) 2 = εe ε κ log κ 1 - a 0 log κ + O J κ (log κ) J+3 . (2.5.16) Writing g(K, τ ) -g(κ, τ ) = g(K, T) -g(κ, τ ) + g(K, τ ) -g(K, T)
and using (2.5.15) and (2.5.16), we can derive that

g(K, τ ) -g(k, τ ) -(a 1 -a 2 0 -C 1 )(e ε -1) + C 0 εe ε κ (log κ) 2 -(e ε -1 -ε) κ log κ + O J κ (log κ) J+3 -(a 1 -a 2 0 -C 1 + C 0 ) εκ (log κ) 2 + O J κ (log κ) J+3 = - εκ (log κ) 2 + O J κ (log κ) J+3 , thanks to the choice of a 1 = a 2 0 + C 1 -C 0 + 1 = C 2 0 + C 0 + C 1 + 2.
Thus the inequality (2.5.11) can be written as

∞ τ +ε Φ T (t; y)t 2κ-1 dt ∞ 0 Φ T (t; y)t 2κ-1 dt exp - 2εκ (log κ) 2 + O J κ 2 y + κ (log κ) J+3 for τ 0 τ log 2 T -20 and κ = κ τ . This implies that ∞ τ +ε Φ T (t; y)t 2κ-1 dt 1 4 ∞ 0 Φ T (t; y)t 2κ-1 dt, (2.5.17)
provided the constant c is suitably large and y κ(log κ) J+3 . Similarly

τ -ε 0 Φ T (t; y)t 2κ-1 dt 1 4 ∞ 0 Φ T (t; y)t 2κ-1 dt. (2.5.18)
From (2.5.17) and (2.5.18), we deduce that

1 2 ∞ 0 Φ T (t; y)t 2κ-1 dt τ +ε τ -ε Φ T (t; y)t 2κ-1 dt ∞ 0 Φ T (t; y)t 2κ-1 dt.
Combining this with Lemma 2.5.2 leads to

τ +ε τ -ε Φ T (t; y)t 2κ-1 dt = (log κ) 2κ 2κ exp 2κ log κ J j=0 C j (log κ) j + O J κ y + 1 (log κ) J+1
(2.5. [START_REF] Cheng | An explicit upper bound for the Riemann zeta-function near the line σ = 1[END_REF] uniformly for (T, y, κ) in (2.5.3) above and (2.5.4).

On the other hand, in view of the fact that Φ T (t; y) is decreasing in t, we have

Φ T (τ + ε; y)(τ -ε) 2κ-1 2k τ +ε τ -ε Φ T (t; y)t 2κ-1 dt Φ T (τ -ε; y)(τ + ε) 2κ-1 (2.5.20)
Since τ = log κ + J+1 j=0 a j /(log κ) j and ε (log κ) -J-1 , it follows that

τ ± ε log κ -2k τ ± ε 2κ = exp -2κ log 1 + J j=0 a j (log κ) j+1 + O 1 (log κ) J+3 = exp - 2κ log κ a 0 + J j=1 a j (log κ) j + O 1 (log κ) J+1
.

(2.5.21)

where the a j are constants ( a 1 = a 1 + a 2 0 ). From (2.5.19)-(2.5.21), we can deduce that

Φ T (τ + ε; y) exp - 2κ log κ 2 + J j=1 a j -C j (log κ) j + O J κ y + 1 (log κ) J+1
Φ T (τ -ε; y).

Since τ = log κ + J j=0 a j /(log κ) j , we can apply Lemma 2.4.2 to write

2κ log κ = 2e τ -C 0 -1+ J j=1 b j /τ j τ + J j=0 b j /τ j = 2e τ -C 0 -1 τ 1 + J j=1 b j τ j + O 1 τ J+1 and J j=1 a j -C j (log κ) j = J j=1 a j -C j (τ + J =0 b /τ ) j = J j=1 c j τ j + O 1 τ J+1 . Combining (2.5.19)-(2.5.21), we obtain Φ T (τ + ε; y) exp - 2e τ -C 0 -1 τ 1 + J j=1 a j τ j + O J 1 τ J+1 + e τ y Φ T (τ -ε; y) (2.5.22)
with

a 1 = 2b 1 + c 1 = 2(b 1 -b 0 ) + a 1 -C 1 = 2(-a 1 + a 0 ) + a 1 + a 2 0 -C 1 = 2a 0 -a 1 + a 2 0 -C 1 = 2a 0 -(a 2 0 + C 1 ) + a 2 0 -C 1 = 2a 0 -2C 1 = 2(1 + C 0 -C 1 ).
Φ T (τ ) = exp - 2e τ -C 0 -1 τ 1 + J j=1 a j τ j + O J 1 τ J+1 + log T y + O exp - log T 50 log 2 T .
This implies the required result by choosing y = min{(log T )τ 2J+2 , (log T ) 2 /e 10+τ }.

3 Distribution of |ζ(σ + it)| in the strip 1 2 < σ < 1

Background

Throughout this section, σ will denote any fixed number in ( 1 2 , 1), ζ(s) the Riemann zeta function and log j the j-th iterated logarithm. Firstly we make a brief review of the extreme values of |ζ(σ + it)| as t varies. In 1928, Titchmarsh [START_REF] Titchmarsh | On an inequality satisfied by the zeta-function of Riemann[END_REF] showed that for any ε > 0, we have lim sup

t→∞ log |ζ(σ + it)| (log t) 1-σ-ε = ∞.
In 1972, Levinson [START_REF] Levinson | Ω-theorems for the Riemann zeta-function[END_REF] replaced (log t) ε by log 2 t, by showing that for sufficiently large T we have max

t∈[0,T ] log |ζ(σ + it)| (log T ) 1-σ log 2 T .
In 1977, Montgomery [START_REF] Montgomery | Extreme values of the Riemann zeta function[END_REF] showed that max

t∈[0,T ] log |ζ(σ + it)| ν(σ) (log T ) 1-σ (log 2 T ) σ , (3.1.1)
where ν(σ) = 1 20 (σ -1 2 )

1 2 unconditionally, and ν(σ) = 1 20 on assuming the Riemann hypothesis. This quantity (log T ) 1-σ /(log 2 T ) σ is conjectured to be the true order of magnitude of max t∈[0,T ] log |ζ(σ + it)|. More precisely, we believe the following inequality holds:

max t∈[0,T ] log |ζ(σ + it)| σ (log T ) 1-σ (log 2 T ) σ •
Thus, the only improvement of (3.1.1) we could expect is to get larger values of ν(σ). We refer to [START_REF] Aistleitner | Lower bounds for the maximum of the Riemann zeta function along vertical lines[END_REF][START_REF] Bondarenko | Note on the resonance method for the Riemann zeta function, 50 years with Hardy spaces[END_REF].

In 2011, applying a method of Granville and Soundararajan [START_REF] Granville | The Riemann Zeta Functionn and Related Themes: Papers in Honour of Professor K. Ramachandra[END_REF] to investigate the distribution of values of |ζ(1 + it)|, Lamzouri [START_REF] Lamzouri | On the distribution of extreme values of Zeta and L-functions in the stip 1 2 < σ < 1[END_REF] Then he [START_REF] Lamzouri | On the distribution of extreme values of Zeta and L-functions in the stip 1 2 < σ < 1[END_REF] showed that there exists a positive constant c(σ) such that

Φ T (τ ) = exp -(τ log σ τ ) 1 1-σ a 0 + O 1 √ log τ + (τ log τ ) 1 1-σ log T σ-1 2 (3.1.3) uniformly in the range 1 τ c(σ)(log T ) 1-σ / log 2 T
, where a 0 will be defined later in (3.1.4). Despite the maximum of the range of τ being much less than (3.1.1), the distribution function (3.1.3) has more significance. If (3.1.3) were to persist to the end of the viable range, then we could get a conjectural value of max t∈[T,2T ] log |ζ(σ + it)|. More precisely, we have Lamzouri's conjecture (see [START_REF] Lamzouri | On the distribution of extreme values of Zeta and L-functions in the stip 1 2 < σ < 1[END_REF]):

max t∈[T,2T ] log |ζ(σ + it)| = {c(σ) + o(1)} (log T ) 1-σ (log 2 T ) σ holds for T → ∞, where c(σ) := C 0 σ 2σ (1 -σ) 1-σ
and C 0 will be defined in (3.3.2). Note that this conjecture also implies the upper bound of |ζ(σ + it)|. For more work concerning it, we refer to [START_REF] Chandee | Bounding |ζ( 1 2 + it)| on the Riemann Hypothesis[END_REF][START_REF] Cheng | An explicit upper bound for the Riemann zeta-function near the line σ = 1[END_REF][START_REF] Ford | Vinogradov's integral and bounds for the Riemann zeta function[END_REF][START_REF] Richert | Zur Abschätzung der Riemannschen Zetafunktion in der Nähe der Vertikalen σ = 1[END_REF][START_REF] Titchmarsh | The theory of the Riemann zeta-function, 2nd edn[END_REF].

In this section, we aim to improve the asymptotic distribution function (3.1.3). We have a higher order expansion in the exponent, which is inspired by the work in [START_REF] Wu | Note on a paper by A[END_REF].

Theorem 3.1.1. Let σ ∈ ( 1 2 , 1
) be a fixed real number. Let Φ T (τ ) be defined in (3.1.2). Then there exists a sequence of polynomials with real coefficients {a n (•)} n 0 with deg(a n ) n, and a constant c(σ) > 0, such that for any integer N 1, we have

Φ T (τ ) = exp -(τ log σ τ ) 1 1-σ N n=0 a n (log 2 τ ) (log τ ) n + O log 2 τ log τ N +1 + (τ log τ ) 1 1-σ log T σ-1 2 uniformly for T → ∞ and 1 τ c(σ)(log T ) 1-σ / log 2 T
, where the implied constant depend on N and σ. Especially, we have

a 0 := σ 2σ C σ 0 (1 -σ) 2σ-1 1/(1-σ) (3.1.4)
with C 0 defined in (3.3.2).

The main new ingredient for the proof of Theorem 3.1.1 is Proposition 3.4.1 below, which gives a better approximation of the distribution function of the short Euler products:

Φ T (τ ; y) := 1 T meas t ∈ [T, 2T ] : log |ζ(σ + it; y)| > τ , (3.1.5)
where

ζ(σ + it; y) := p y 1 - 1 p σ+it -1
.

We refer to [START_REF] Mine | Large deviations for values of L-functions attached to cusp forms in the level aspect[END_REF] for similar work on L-functions attached to cusp forms.

Preliminary lemmas

Firstly, we will show the relationship between sums attached to the divisor function and the Bessel function by two asymptotic formulas. These will be used in the progress of calculating the moments of the short Euler products for the Riemann zeta function and the Dirichlet L-functions. One should pay attention that here k is not necessarily an integer.

The modified Bessel function I 0 (t) of order 0 is defined by

I 0 (t) := 1 0 exp(t cos(2πθ)) dθ = ∞ n=0 (t/2) 2n
(n!)

d k/2 (p ν ) 2 p 2νσ = I 0 k p σ exp O σ k p 2σ , (3.2.5) ν 0 d k/2 (p ν ) 2 p 2νσ = exp O σ k p σ (p k 1/σ ), (3.2.6)
where the implied constants depend on σ only.

Proof. See also of [START_REF] Granville | The Riemann Zeta Functionn and Related Themes: Papers in Honour of Professor K. Ramachandra[END_REF]Lemma 4]. Writing e(θ) := e 2πiθ , then

1 - e(θ) p σ -k = 1 - e(θ) p σ -k/2 1 - e(-θ) p σ -k/2 = ν 0 ν 0 d k/2 (p ν )d k/2 (p ν )e((ν -ν )θ) p (ν+ν )σ • Thus we can derive that ν 0 d k/2 (p ν ) 2 p 2νσ = 1 0 1 - e(θ) p σ -2(k/2) dθ = 1 0 1 - 2 cos(2πθ) p σ + 1 p 2σ -k/2 dθ = 1 0 exp - k 2 log 1 - 2 cos(2πθ) p σ + 1 p 2σ dθ.
This implies (3.2.5) thanks to the formula log(1

+ t) = t + O(t 2 ) (|t| 2 -1 2 ),
1 p σ = x 1-σ (1 -σ) log x + O x 1-σ (1 -σ) 2 (log x) 2
uniformly for x → ∞ and 1 2 < σ < 1, where the implied constant is absolute.

Choosing k = (4e -1 (σ -1 2 )λ 2 y 2σ-1 log y , which satisfies the condition in Lemma 3.2.5, then by this lemma we have 1

T 2T T y p z 1 p σ+it 2k dt k y p z 1 p 2σ k + 1 T 1 3 k (σ -1 2 )y 2σ-1 log y k + 1 T 1 3 • So the frequency of t ∈ [T, 2T ] such that | log ζ(σ + it; z) -log ζ(σ + it; y)| > 2λ is less than 1 T 2T T 1 2λ y p z 1 p σ+it 2k dt k 4(σ -1 2 )λ 2 y 2σ-1 log y k + (2λ) -2k T -1 3 . (3.2.9) Since λ > c 1 (σ)(log 2 T / log T ) 2 , we have | log ζ(σ + it) -log ζ(σ + it; y)| | log ζ(σ + it; z) -log ζ(σ + it; y)| -| log ζ(σ + it; z) -log ζ(σ + it)| 2λ + O((log 2 ) 2 /(log T ) 2 ) > λ.
By (3.2.8) and (3.2.9), the frequency of t

∈ [T, 2T ] such that | log ζ(σ + it) -log ζ(σ + it; y)| > 2λ is less than, thanks to our choice of k, k 4(σ -1 2 )λ 2 y 2σ-1 log y k + 1 (2λ) 2k T 1 3 + 1 T 1 4 (σ-1 2 ) e -k + (2λ) -2k T -1 4 (σ-1 2 ) .
This implies the required result, since our hypothesis on (λ, y) garanties

(2λ) -2k T -1 4 (σ-1 2 ) T -1 8 (σ-1 2 )
exp(-4e -1 (σ -1 2 )λ 2 y 2σ-1 log y). Combining this with the first step, Lemma 3.2.6 follows.

Moments of the short Euler products

In this section, we will evaluate the k-th moment of the short Euler product ζ(σ + it; y) by proving the following proposition, which is important for the proof of Theorem 3.1.1. It has a higher order expansion in the exponent, which is an improvement of equation (4.2) in [START_REF] Lamzouri | On the distribution of extreme values of Zeta and L-functions in the stip 1 2 < σ < 1[END_REF].

Proposition 3.3.1. Let σ ∈ ( 1 2 , 1
) be a fixed constant and let N be a non-negative integer. Then we have

1 T 2T T |ζ(σ + it; y)| k dt = exp k 1/σ log k N n=0 C n (log k) n + O 1 (log k) N +1 + k 1/σ y 2σ-1
uniformly for T 3 and

ky 1-σ 1 8 (1 -σ) log T, (3.3.1)
where

C n := ∞ 0 (log t) n t 1/σ+1 log I 0 (t) dt (n 0) (3.3.2)
and I 0 (t) is the Bessel function given by (3.2.1). Especially, we have C 0 > 0.

The integer n 1 is called y-friable if the largest prime factor P (n) of n is less than y (P (1) = 1 by convention). Denote by S(y) the set of y-friable integers. We will show that, in the expansion of the k-th moment of ζ(σ + it; y), the diagonal terms lead to the main term, while the off-diagonal terms contribute to the error term. Again we strengthen that, k is not necessarily an integer.

d k/2 (n) 2 n 2σ = exp k 1/σ log k N n=0 C n (log k) n + O 1 (log k) N +1 + k 1/σ y 2σ-1 . ( 3
d k/2 (n) 2 n 2σ = p y ν 0 d k/2 (p ν ) 2 p 2νσ = exp O p k 1/σ k p σ = exp O k 1/σ log k .
Now we treat the case of y > k

d k/2 (n) 2 n 2σ = p k 1/(2σ) exp O k p σ k 1/(2σ) <p y I 0 k p σ exp O k p 2σ = exp O σ,N k 1/σ (log k) N +2 k 1/(2σ) <p y I 0 k p σ , (3.3.4)
where the last equation holds since

p k 1/(2σ) k p σ k (k 1/(2σ) ) 1-σ log k 1/(2σ) k 1 2 +1/(2σ) log k σ,N k 1/σ (log k) N +2 and k 1/(2σ) <p y k p 2σ k 1/(2σ) log k σ,N k 1/σ (log k) N +2 •
Next we evaluate the second factor on the right-hand side de (3.3.4). Taking the logarithm of this factor and using the prime number theorem, we have log

k 1/(2σ) <p y I 0 k p σ = y k 1/(2σ) log I 0 k u σ dπ(u) = M + E, (3.3.5)
where

M := y k 1/(2σ) log I 0 k u σ du log u , E := y k 1/(2σ) log I 0 k u σ dO ue -c √ log u .
In view of (3.2.2) and (3.2.3), we always have log I 0 (t) t 2 (t 0). Thus using this bound and (3.2.4), we can derive that

E = log I 0 k u σ O ue -c √ log u y k 1/(2σ) - y k 1 2 σ log I 0 k u σ O ue -c √ log u du k y σ 2 y e c √ log y + k 1 2 +1/(2σ) e c √ log k + k k 1/σ k 1/(2σ) e -c √ log u u σ du + k 2 y k 1/σ e -c √ log u u 2σ du k 1/σ y 2σ-1 k 1/σ e -c √ log y + k 1 2 +1/(2σ) e -c √ log k + k 1/σ e -c √ log k k 1/σ log k k 1/σ y 2σ-1 log k e c √ log y + k -(1/σ-1)/2 log k e c √ log k . (3.3.6)
This is acceptable, since y k 1/σ . In order to calculate the main term of (3.3.5), setting t = k/u σ , and integrating by substitution, then we have

M = k 1/σ k 1 2 k/y σ log I 0 (t) t 1/σ+1 log(k/t) dt = k 1/σ log k k 1 2 k/y σ log I 0 (t) t 1/σ+1 1 1 -log t/ log k dt.
For k/y σ t k 1 2 , we can write

1 1 -log t/ log k = N n=0 (log t) n (log k) n + O σ,N (log t) N +1 (log k) N +1 . Thus M = k 1/σ log k N n=0 C n (k, y) (log k) n + O 1 (log k) N +1
, where k/y σ (log t) N +1 t 1/σ+1 log I 0 (t) dt

C n (k, y) := k 1 2 k/y σ (log t) n t 1/
1 k/y σ (log t) N +1 t 1/σ-1 dt + k 1 2 1 (log t) N +1 t 1/σ dt σ,N 1.
On the other hand, we enlarge the integral interval to (0, ∞), and use the definition of C n , then the main term of the last formula is

C n (k, y) = C n -C n where C n := k/y σ 0 + ∞ k 1 2 (log t) n t 1/σ+1 log I 0 (t)dt k/y σ 0 (-log t) n t 1/σ-1 dt + ∞ k 1 2 (log t) n t 1/σ dt σ,N k 1/σ y 2σ-1 (log(y σ /k)) n + (log k) n k (1/σ-1)/2 , thanks to (3.2.2)-(3.2.3). It follows that C n (log k) n k 1/σ y 2σ-1 log(y σ /k) log k n + 1 k (1/σ-1)/2 k 1/σ y 2σ-1 + 1 (log k) N +1 , since log(y σ /k) log k n σ,N 1 if y k 2/σ
and otherwise we have

k 1/σ y 2σ-1 log(y σ /k) log k n k 1/σ k 2/σ 2σ-1 log((k 2/σ ) σ /k) log k n 1 (log k) N +1 • Thus M = k 1/σ log k N n=0 C n (log k) n + O σ,N 1 (log k) N +1 + k 1/σ y 2σ-1 . ( 3

Proof of Theorem 3.1.1

Recall that we have define the short Euler products by

ζ(σ + it; y) := p y 1 - 1 p σ+it -1
, and its distribution function

Φ T (τ ; y) := 1 T meas t ∈ [T, 2T ] : log |ζ(σ + it; y)| > τ .
In this section, we aim to prove the following proposition.

Proposition 3.4.1. Let σ ∈ ( 1 2 , 1) be a fixed constant and let N 1 be an integer and let c 0 = c 0 (σ, N ) be a large positive constant depending on (σ, N ). Then we have

Φ T ((1 + ε 0 )τ ; y) exp -(τ log σ τ ) 1 1-σ N n=0 a n (log 2 τ ) (log τ ) n + O(ε 0 ) Φ T ((1 -ε 0 )τ ; y) uniformly for T → ∞, log T y (log T ) 2 , 1 τ c(σ) log 2 T log T y 1-σ 1-σ σ , (3.4.1)
where c(σ) is a positive constant depending only on σ,

ε 0 = ε 0 (τ, y) = c 0 log 2 τ log τ N +1 + (τ log τ ) 1 1-σ y σ-1 2 , (3.4.2)
the polynomials a n (•) is the same as in Theorem 3.1.1 and the implied constant is absolute.

Two preliminary lemmas

The following lemma relates the moments of the short Euler products to the distribution function.

Lemma 3.4.2. Let σ ∈ ( 1 2 , 1) be a fixed constant. For any non-negative integer N , we have

∞ -∞ Φ T (t; y)ke kt dt = exp k 1/σ log k N n=0 C n (log k) n + O 1 (log k) N +1 + k 1/σ y 2σ-1
uniformly for (T, y, k) in (3. Lemma 3.4.3. Let σ ∈ ( 1 2 , 1) be a fixed constant. Let {a n } n 0 be a sequence of real numbers and N 0 be an integer. If

τ = k 1/σ-1 σ log k N +1 n=0 a n (log k) n (k → ∞), (3.4.3)
then there is a sequence of polynomials

{b n (•)} n 0 with deg(b n ) n and b 0 = σ 1-σ such that log k = (log τ ) N n=0 b n (log 2 τ ) (log τ ) n + O log 2 τ log τ N +1 , (3.4.4)
where the implied constant depends on the sequence {a n } n 0 and N .

Proof. We prove it by recurrence. Firstly, taking logarithm of both sides in (3.4.3), we have

log τ = 1 -σ σ log k -log σ -log 2 k + log N +1 n=0 a n (log k) n . (3.4.5)
From this we derive that

log(τ log τ ) = 1 -σ σ log k + O σ (1)
and

log k = σ 1 -σ log(τ log τ ) + O σ (1) = σ 1 -σ (log τ ) 1 + log 2 τ + O σ (1) log τ , (3.4.6)
which is the case for N = 0. Now assume we already have

log k = (log τ ) m n=0 b n (log 2 τ ) (log τ ) n + O log 2 τ log τ m+1 ,
for some m < N . Inserting this into (3.4.5), it follows that

log τ = 1 -σ σ log k -log σ -log (log τ ) m n=0 b n (log 2 τ ) (log τ ) n + O log 2 τ log τ m+1 + log N +1 n=0 a n (log τ ) m n=0 b n (log 2 τ ) (log τ ) n + O log 2 τ log τ m+1 -n
, from which we derive that

1 -σ σ log k = (log τ ) 1 + log σ + log 2 τ log τ + 1 log τ log m n=0 b n (log 2 τ ) (log τ ) n + O log 2 τ log τ m+1 - 1 log τ log N +1 n=0 a n (log τ ) m n=0 b n (log 2 τ ) (log τ ) n + O log 2 τ log τ m+1 -n
.

By expansion of the log-terms, we can obtain

1 -σ σ log k = (log τ ) m+1 n=0 b * n (log 2 τ ) (log τ ) n + O log 2 τ log τ m+2 with some polynomials b * n (log 2 τ ) of deg(b * n ) n

Proof of Proposition 3.4.1

Let {a n } n 0 be a real sequence depending on σ, which will be chosen later. It is clear that there is a large constant t 0 = t 0 (σ) such that the function

t → t 1/σ-1 σ log t N +1 n=0 a n (log t) n
is strictly increasing on [t 0 , ∞). Thus we choose a unique k such that

τ = k 1/σ-1 σ log k N +1 n=0 a n (log k) n • (3.4.7)
Noticing that (3.4.6) and (3.4.1) imply that

ky 1-σ σ (τ log τ ) σ 1-σ y 1-σ σ log T y 1-σ y 1-σ 1 8 (1 -σ) log T,
we can apply Lemma 3.4.2 to write

∞ -∞ Φ T (t; y)ke kt dt = exp k 1/σ log k 2N +1 n=0 C n (log k) n + O σ,N (R 2N +2 (k, y)) , (3.4.8)
where

R 2N +2 (k, y) := 1 (log k) 2N +2 + k 1/σ y 2σ-1
.

We choose

ε = A 1 (log k) N +1 + k 1/σ y σ-1 2 ∈ (0, 10 -2022 ) (k k 0 ), (3.4.9)
where A = A(σ, N ) and k 0 = k 0 (σ, N ) are large constants depending on (σ, N ), and let

k 1 := (1 + ε)k, k 2 := (1 -ε)k, τ 1 := 1 + ε 2σ τ, τ 2 := 1 - ε 2σ τ.
When t τ 2 , we have

kt (k -k 2 )(τ 2 -t) + kt = (k -k 2 )τ 2 + k 2 t = εkτ 2 + k 2 t. Thus τ 2 -∞ e kt Φ T (t; y)dt e εkτ 2 ∞ -∞ e k 2 t Φ T (t; y)dt. (3.4.10)
Using (3.4.8) and noticing that

R 2N +2 (k 2 , y) σ,N R 2N +2 (k, y), we have ∞ -∞ e k 2 t Φ T (t; y)dt = exp k 1/σ 2 log k 2 2N +1 n=0 C n (log k 2 ) n + O σ,N (R 2N +2 (k, y)) .
Inserting this into (3.4.10) and using the definition of τ 2 with (3.4.7), then we have

τ 2 -∞ e kt Φ T (t; y)dt exp k 1/σ log k {S 1 + S 2 + O σ,N (R 2N +2 (k, y))} , (3.4.11)
where

S 1 := ε σ 1 - ε 2σ N +1 n=0 a n (log k) n , S 2 := (1 -ε) 1/σ 1 + log(1 -ε)/ log k 2N +1 n=0 C n (log k) n 1 (1 + log(1 -ε)/ log k) n •
The first part S 1 can be calculated easily, using the choice of ε, as

S 1 = ε σ N +1 n=0 a n (log k) n - ε 2 2σ 2 a 0 + o σ,N (R 2N +2 (k, y)).
(3.4.12)

In order to calculate the second part S 2 , we take Taylor series for log(1-ε), use the geometric series formula, and put all infinitesimal of higher order than R 2N +2 into the error term, then we have

S 2 = 1 - ε σ + ε 2 2σ 1 σ -1 1 + ε log k 2N +1 n=0 C n (log k) n 1 + nε log k + o σ,N (R 2N +2 (k, y)) = 1 - ε σ + ε log k + ε 2 2σ 2 - ε 2 2σ 2N +1 n=0 C n (log k) n 1 + nε log k + o σ,N (R 2N +2 (k, y)).
We separate the same part as in the exponent of (3.4.12) from the above formula, and again put all the infinitesimal of higher order than R 2N +2 into the error term, then we can write

S 2 = 2N +1 n=0 C n (log k) n + ε 2 2σ 2 C 0 - ε 2 2σ C 0 + - ε σ + ε σ C 0 N +1 n=1 σnC n-1 -C n (log k) n + o σ,N (R 2N +2 (k, y)).
(3.4.13) Combining (3.4.12) and (3.4.13), we have

S 1 + S 2 = 2N +1 n=0 C n (log k) n + ε σ N +1 n=1 a n + σnC n-1 -C n (log k) n + ε σ - ε 2 2σ 2 (a 0 -C 0 ) - ε 2 2σ C 0 + o σ,N (R 2N +2 (k, y)).
Choosing a 0 = C 0 and a n = C n -σnC n-1 for n 1, we find that

S 1 + S 2 = 2N +1 n=0 C n (log k) n - ε 2 2σ C 0 + o σ,N (R 2N +2 (k, y)).
Inserting this into (3.4.11), and using (3.4.8), then we have

τ 2 -∞ e kt Φ T (t; y)dt exp k 1/σ log k - ε 2 2σ C 0 + 2N +1 n=0 C n (log k) n + O σ,N (R 2N +2 (k, y)) = exp k 1/σ log k - ε 2 2σ C 0 + O σ,N (R 2N +2 (k, y)) ∞ -∞
e kt Φ T (t; y)dt.

By the choice of the value of ε (A = A(σ, N ) is a suitably large constant), and C 0 > 0, we can obtain

τ 2 -∞ e kt Φ T (t; y)dt 1 4 ∞ -∞
e kt Φ T (t; y)dt.

Similarly, we have

∞ τ 1 e kt Φ T (t; y)dt 1 4 ∞ -∞
e kt Φ T (t; y)dt.

Thus combining the above two inequalities we have

1 2 ∞ -∞ e kt Φ T (t; y)dt τ 1 τ 2 e kt Φ T (t; y)dt ∞ -∞
e kt Φ T (t; y)dt.

So thanks to (3.4.8), we can get the asymptotic formula for the integral over (τ 2 , τ 1 ):

τ 1 τ 2 e kt Φ T (t; y)dt = exp k 1/σ log k 2N +1 n=0 C n (log k) n + O(R 2N +2 (k, y)) . (3.4.14)
On the other hand, since Φ T (t; y) is decreasing in t, we have

(τ 1 -τ 2 )e kτ 2 Φ T (τ 1 ; y) τ 1 τ 2 e kt Φ T (t; y)dt (τ 1 -τ 2 )e kτ 1 Φ T (τ 2 ; y).
By the choice of the values of τ 1 and τ 2 , the above inequality is

ετ σ e kτ (1-ε 2σ ) Φ T ((1 + ε 2σ )τ ; y) τ 1 τ 2 e kt Φ T (t; y)dt ετ σ e kτ (1+ ε 2σ ) Φ T ((1 -ε 2σ )τ ; y). (3.4.15)
In view of (3.4.7), it is easy to see that

σ ετ e -kτ (1± ε 2σ ) = exp log σ ετ -kτ {1 + O(ε)} = exp - k 1/σ σ log k N +1 n=0 a n (log k) n + O(ε) .
Combining this with (3.4.14) and (3.4.15), it follows that

σ ετ e -kτ (1± ε 2σ ) τ 1 τ 2 e kt Φ T (t; y)dt = exp - k 1/σ σ log k N n=0 a n -σC n (log k) n + O(ε) .
Back to (3.4.15), we get

Φ T ((1 + ε 2σ )τ ; y) exp - k 1/σ σ log k N n=0 a n -σC n (log k) n + O(ε) Φ T ((1 -ε 2σ )τ ; y). (3.4.16)
Recall that Lemma 3.4.3 and (3.4.6) give

log k = log τ N n=0 b n (log 2 τ ) (log τ ) n + O log 2 τ log τ N +1 and log k = σ 1 -σ log(τ log τ ) + O σ (1).
With the help of these formulas, after some computations of Taylor's expansions we easily see that there are a sequence of polynomials {a n (•)} n 0 * with deg(a n ) n and a positive constant c 0 = c 0 (σ, N ) depending on (σ, N ) such that

k 1/σ σ log k N n=0 a n -σC n (log k) n + O(ε) = (τ log σ τ ) 1 1-σ N n=0 a n (log 2 τ ) (log τ ) n + O(ε 0 ) (3.4.17) and ε/(2σ) ε 0 , (3.4.18)
where ε 0 is given as in (3.4.2). Inserting (3.4.17) into (3.4.16) and using the fact that the function t → Φ T (t; y) is decreasing with (3.4.18), we obtain the required result. This completes the proof.

End of the proof of Theorem 3.1.1

Let

η := c 0 (τ log τ ) 1 1-σ y σ-1 2
, where c 0 = c 0 (σ, N ) be a large positive constant given as in Proposition 3.4.1. Applying Lemma 3.2.6 with λ = ητ , we can obtain

Φ T (τ ) = Φ T (τ (1 ± η); y) + O exp -(4e) -1 (σ -1 2 )c 2 0 log y log τ (τ log σ τ ) 1 1-σ . (3.4.19)
On the other hand, noticing that η ε 0 and that Φ T (t; y) is decreasing in t, (3.4.19) and Proposition 3.4.1 imply that

Φ T (τ ) = exp -(τ log σ τ ) 1 1-σ N n=0 a n (log 2 τ ) (log τ ) n + O(ε 0 ) + O exp -(4e) -1 (σ -1 2 )c 2 0 log y log τ (τ log σ τ ) 1 1-σ = exp -(τ log σ τ ) 1 1-σ N n=0 a n (log 2 τ ) (log τ ) n + O(ε 0 ) ∆(τ, y), (3.4.20)
uniformly for

T → ∞, log T y (log T ) 2 , 1 τ c(σ) log 2 T log T y 1-σ 1-σ σ , where ∆(τ, y) := 1 + O exp -(4e) -1 (σ -1 2 )c 2 0 log y log τ (τ log σ τ ) 1 1-σ + O σ,N (τ log σ τ ) 1 1-σ .
Since c 0 is suitably large, we have, with choice of y = log T ,

∆(τ, log T ) = 1 + O exp -(8e) -1 (σ -1 2 )c 2 0 (τ log σ τ ) 1 1-σ * The value of a 0 follows easily from b 0 = σ 1-σ in Lemma 3.4.3.

Large values of |ζ(σ + it)|

Background

In this section, we investigate the extreme values of the Riemann zeta function ζ(s) in the strip 1 2 < e s 1. The study of the values on the 1-line can date back to 1925 when Littlewood [START_REF] Littlewood | On the Riemann zeta-function[END_REF] showed that there exists arbitrarily large t for which

|ζ(1 + it)| (1 + o(1))e γ log 2 t.
Here and throughout, we denote by log j the j-th iterated logarithm and by γ the Euler constant. This was improved by Levinson [START_REF] Levinson | Ω-theorems for the Riemann zeta-function[END_REF], who in 1972 proved that there exists arbitrarily large t such that |ζ(1 + it)| e γ log 2 t + O(1).

In 2006, Granville and Soundararajan [START_REF] Granville | The Riemann Zeta Functionn and Related Themes: Papers in Honour of Professor K. Ramachandra[END_REF] used Diophantine approximation to prove that the lower bound for some constant C 0 = -0.3953997, which provides a precise description of the extreme values. For the upper bound, the best unconditional result is established by Vinogradov [START_REF] Vinogradov | A new estimate of the function ζ(1 + it)[END_REF] who proved that |ζ(1 + it)| (log t) 2 3 . In recent years, the resonance method has been extensively developed, which can detect extreme values of the Riemann zeta function more effectively. It was first used by Voronin [START_REF] Voronin | Lower bounds in Riemann zeta-function theory Izv[END_REF] in 1988, and developed by Soundararajan [START_REF] Soundararajan | Extreme values of zeta and L-functions[END_REF] in 2008 and Hilberdink [START_REF] Hilberdink | An arithmetical mapping and applications to Ω-results for the Riemann zeta function[END_REF] Note that this requires a larger range [ √ T , T ] than [T, 2T ] in (4.1.1), which is typical in the application of "long resonance". Their method can also apply to a class of generalized L-functions (see [START_REF] Dixit | Large values of L-functions on the 1-line[END_REF]). Inspired by their work, the aim of this section is to get an improved lower bound of large values, which presents an explicit description of the error term O(1) . For any β ∈ (0, 1), we define

Z β (σ, T ) := max T β |t| T |ζ(σ + it)|.
Then we have the following theorem. For any fixed σ ∈ ( 1 2 , 1), Titchmarsh [START_REF] Titchmarsh | On an inequality satisfied by the zeta-function of Riemann[END_REF] in 1928 showed that for any ε > 0 there exists arbitrarily large t such that log |ζ(σ + it)| (log t) 1-σ-ε .

In 1972, Levinson [34] improved this result by showing that for large T there exists a positive c such that max

t∈[0,T ] log |ζ(σ + it)| c (log T ) 1-σ log 2 T .
In 1977, Montgomery [START_REF] Montgomery | Extreme values of the Riemann zeta function[END_REF] showed that max

t∈[0,T ] log |ζ(σ + it)| ν(σ) (log T ) 1-σ (log 2 T ) σ , (4.1.3)
where ν(σ) = 1 20 σ -1 2 unconditionally, and ν(σ) = 1 20 under the Riemann hypothesis. In 2016, using the resonance method, Aistleitner [START_REF] Aistleitner | Lower bounds for the maximum of the Riemann zeta function along vertical lines[END_REF] improved Montgomery's unconditional result by showing that (4.1.3) holds for ν(σ) = 0.18(2σ -1) 1-σ .

For the upper bound, Richert [START_REF] Richert | Zur Abschätzung der Riemannschen Zetafunktion in der Nähe der Vertikalen σ = 1[END_REF] in 1967 proved that

|ζ(σ + it)| At B(1-σ) 3/2 (log t) 2/3 = At B(1-σ) 3/2 +ε
holds for some absolute A and B. Successive improvements of Richert's bound have reduced the admissible size of A and B. We refer to Cheng [START_REF] Cheng | An explicit upper bound for the Riemann zeta-function near the line σ = 1[END_REF] and Ford [START_REF] Ford | Vinogradov's integral and bounds for the Riemann zeta function[END_REF] for more details. Under the Riemann Hypothesis, we could have a much better upper bound (see [START_REF] Lamzouri | On the distribution of extreme values of Zeta and L-functions in the stip 1 2 < σ < 1[END_REF] and [START_REF] Titchmarsh | The theory of the Riemann zeta-function, 2nd edn[END_REF])

log |ζ(σ + it)| (log t) 2-2σ log 2 t . (4.1.4)
It is conjectured that the true order of the magnitude of max t∈[0,T ] log |ζ(σ + it)| corresponds to the lower bound (4.1.3) rather than the upper bound (4.1.4). In 2011, based on the probabilistic model, Lamzouri [START_REF] Lamzouri | On the distribution of extreme values of Zeta and L-functions in the stip 1 2 < σ < 1[END_REF] gave an explicit conjectural value of c(σ), claiming that max

t∈[0,T ] log |ζ(σ + it)| = c(σ) (log T ) 1-σ (log 2 T ) σ , holds for c(σ) = 1 σ 2σ (1 -σ) 1-σ ∞ 0 log I 0 (t) t 1/σ+1 dt, (4.1.5)
where I 0 is the modified Bessel function of order 0. In 2018, Bondarenko and Seip [START_REF] Bondarenko | Note on the resonance method for the Riemann zeta function, 50 years with Hardy spaces[END_REF] made celebrated improvement on this topic. They proved that there exists a function ν(σ) which is bounded below by 1/(2 -2σ) and has the asymptotic behavior

ν(σ) = ( √ 2 2 + o(1)) | log(σ -1 2 )| σ 1 2 , (1 -σ) -1 + O(| log(1 -σ)|) σ 1. ( 4
log |ζ(σ + it)| log 3 T + c + ν(σ) (log T ) 1-σ (log 2 T ) σ ,
where c is an absolute constant. In this section, we aim to improve their first result (4.1.7). This is accomplished by deriving a lower bound for the maximum of Gál-type sums, which is a kind of certain greatest common divisor (GCD) sums of the form

Γ σ (N ) exp (2 √ 2 + o(1)) | log(σ -1 2 )| (log N ) 1-σ (log 2 N ) σ .
The following theorem sets up the relation between large values of the Riemann zeta function and the Gál-type sums.

Theorem 4.1.3. For 0 < β < 1 and σ ∈ ( 1 2 + 1 log 2 T , 1), we have

Z β (σ, T ) Γ σ (T 1-β ),
where the implied constant is absolute.

As a direct deduction of Theorem 4.1.2, we get the following lower bound for the Riemann zeta function.

log |ζ(σ + it)| ν β (σ) (log T ) 1-σ (log 2 T ) σ
holds for a function ν β (σ) which has the asymptotic behavior

ν β (σ) = ( √ 2 + o(1))(1 -β) 1-σ | log(σ -1 2 )|, as σ 1 2
.

Therefore when β = 1 2 and σ 1 2 , we improved the result of Bondarenko and Seip by a factor 2 σ . It is also worthy noting that Lamzouri's conjecture (4.1.5) predicts c(σ) ∼ (2σ -1) -1 2 as σ 1 2 . For the sake of completeness, we also mention the large values of the Riemann zeta function on the critical line. For the lower bound, de la Bretèche and Tenenbaum [START_REF] De La Bretèche | Sommes de Gál et applications[END_REF] have shown that

max t∈[0,T ] |ζ( 1 2 + it)| exp ( √ 2 + o(1)) log T log 3 T log 2 T ,
improving earlier results made by Bondarenko and Seip [START_REF] Bondarenko | Large greatest common divisor sums and extreme values of the Riemann zeta function[END_REF][START_REF] Bondarenko | Extreme values of the Riemann zeta function and its argument[END_REF]. For the upper bound, the Lindelöf hypothesis states that for any ε > 0

ζ( 1 2 + it) t ε ,
while the best-known upper bound is due to Bourgain [START_REF] Bourgain | Decoupling, exponential sums and the Riemann zeta function[END_REF] who proved that

ζ( 1 2 + it) t 13 84 +ε .
However, we have the conjectural value due to Farmer, Gonek and Hugh [START_REF] Farmer | The maximum size of L-functions[END_REF] which asserts that max

t∈[0,T ] |ζ( 1 2 + it)| = exp{( √ 2 2 + o(1)) log T log 2 T }.
For much earlier work, we refer to [START_REF] Balasubramanian | On the frequency of Titchmarsh's phenomenon for ζ(s)[END_REF][START_REF] Balasubramanian | On the frequency of Titchmarsh's phenomenon for ζ(s), III[END_REF][START_REF] Chandee | Bounding |ζ( 1 2 + it)| on the Riemann Hypothesis[END_REF][START_REF] Soundararajan | Extreme values of zeta and L-functions[END_REF][START_REF] Titchmarsh | The theory of the Riemann zeta-function, 2nd edn[END_REF]. This chapter is organized as follows. In §4.2, we introduce some preliminary lemmas. We establish the approximation of the Riemann zeta function by its truncated Euler product. In §4.3, we discuss the large values of the Riemann zeta function on 1-line and establish Theorem 4.1.1. In §4.4, we make a brief review on the Gál-type sums and then prove Theorem 4.1.2. Finally in §4.5, we connect Gál-type sums to the values of the Riemann zeta function, and establish Theorem 4.1.3.

Preliminary lemmas

In this section, we introduce some preliminary lemmas. We start with Mertens' formula with an explicit error term. Lemma 4.2.1. Let x > 1000, then we have

1 e γ log x 1 - 1 2(log x) 2 p x 1 - 1 p 1 e γ log x 1 + 1 2(log x) 2 .
Proof. See Theorem 7 of [START_REF] Rosser | Approximate formulas for some functions of prime numbers[END_REF].

The following lemma plays a key role in the proof of Theorem 4.1.2. Lemma 4.2.2. Let x be large, then we have

p x 1 p σ = (1 + o(1)) x 1-σ (1 -σ) log x .
Proof. The is essentially Lemma 6 of [START_REF] Bondarenko | Note on the resonance method for the Riemann zeta function, 50 years with Hardy spaces[END_REF]. The only adjustment lies in the use of prime number theory, where we replace the lower bound by the asymptotic formula

π(x) = (1 + o(1)) x log x .
For analogous statements, see also Lemma 3.1 of [START_REF] Norton | Upper bounds for sums of powers of divisor functions[END_REF], equation (2.1) of [START_REF] Lamzouri | On the distribution of extreme values of Zeta and L-functions in the stip 1 2 < σ < 1[END_REF], and Lemma 3.3 of [START_REF] Bober | The distribution of the maximum of character sums[END_REF].

The following lemma can be seen as a generalization of the greatest common divisors to rational numbers.

σ 1 +iT 0 σ 1 -iT 0 log ζ(1 + it + w) Y w w dw = 2 n Y Λ(n) n 1+it log n + O log T T 0 .
On the other hand, noticing that

p Y, p ν >Y 1 νp ν 1 log Y p Y, ν 2 log p p ν 1 log Y ,
we can write

2 n Y Λ(n) n 1+it log n = p ν Y 1 νp ν(1+it) = p Y ν 1 1 νp ν(1+it) + O 1 log Y = log ζ(1 + it; Y ) + O 1 log Y .
Inserting this into the preceding formula, we get 1 2πi

σ 1 +iT 0 σ 1 -iT 0 log ζ(1 + it + w) Y w w dw = log ζ(1 + it; Y ) + O 1 (log T ) 1/β . (4.2.2)
For the other three integrals, in view of bounds of log ζ(w) in the zero-free region (see [START_REF] Tenenbaum | Introduction to analytic and probabilistic number theory[END_REF]Theorem II.3.16]), typically we have

σ 0 +iT 0 σ 1 +iT 0 + σ 0 -iT 0 σ 0 +iT 0 log ζ(1 + it + w) Y w w dw log T T 0 ,
and

σ 0 -iT 0 σ 0 +iT 0 log ζ(1 + it + w) Y w w dw (log T ) 2 exp(c(log T ) (1/β-1)
) .

Thus the lemma follows from (4.2.1), (4.2.2) and these two bounds. Choose B such that e c+1 < B < (1 -β)/ log 4 and set X := B log T log 2 T . Denote by S(X) the set of all X-friable numbers. Let a n = a(n) be the completely multiplicative function supported on S(X) with a p = 1 -p/X for p X and a p = 0 otherwise. Define the resonator

R(t) = n∈S(X) a n n it = p X (1 -a p p it ) -1 .
Then for t ∈ R we have Set the weight function to be the Gaussian function φ(t) = e -t 2 /2 which satisfies

log |R(t)| p X log(1 -a p ) -1 = π(X) log X -θ(X),
φ(x) = R φ(t)e -itx dt = √ 2πφ(x). Denote M 1 (R; T ) := T β |t| T |R(t)| 2 φ t log T T dt, and
M 2 (R; T ) := T β |t| T ζ(1 + it)|R(t)| 2 φ t log T T dt.
Then clearly we have

Z β (T ) |M 2 (R; T )| M 1 (R; T ) . ( 4

.3.2)

Put Y := exp((log T ) 1/β ) as in Lemma 4.2.4. It follows that

M 2 (R; T ) = 1 + O 1 (log T ) 1/β T β |t| T ζ(1 + it; Y )|R(t)| 2 φ t log T T dt.
Using (4.3.1) and the trivial bound

|ζ(1 + it; Y )| log Y = (log T ) 1/β (t ∈ R),
we can deduce that 1) ,

|t| T β ζ(1 + it; Y )|R(t)| 2 φ t log T T dt T 2B+β+o (
and |t| T ζ(1 + it; Y )|R(t)| 2 φ t log T T dt T 2B+o (1)
∞ T e -(t log T /T ) 2 dt 1.

Thus by denoting 1) .

I 2 (R; T ) = R ζ(1 + it; Y )|R(t)| 2 φ t log T T dt, we have M 2 (R; T ) = 1 + O 1 (log T ) 1/β I 2 (R; T ) + O T 2B+β+o (
On the other hand, we have

M 1 (R; T ) R |R(t)| 2 φ t log T T dt =: I 1 (R; T ),
Therefore by (4.3.2), we derive that

Z β (T ) 1 + O 1 (log T ) 1/β I 2 (R; T ) + O(T 2B+β+o(1) ) I 1 (R; T ) . ( 4

.3.3)

We first estimate the ratio I 2 (R; T )/I 1 (R; T ). For I 1 (R; T ), we have

I 1 (R; T ) = R m,n∈S(X) a m a n m n it φ t log T T dt = T log T m,n∈S(X) a m a n φ T log T log n m φ(0) T log T n∈S(X) a 2 n , (4.3.4)
thanks to the positivity of a n and φ(t). Similarly, for I 2 (R; T ) we have

I 2 (R; T ) = T log T l∈S(Y ) m,n∈S(X) a m a n l φ T log T log nl m .
By sifting the terms with l|m and noticing Y > X, we have that

I 2 (R; T ) T log T l,n,k∈S(X) a kl a n l φ T log T log n k = T log T l∈S(X) a l l k,n∈S(X) a k a n φ T log T log n k .
Therefore, we can deduce that

I 2 (R; T ) I 1 (R; T ) l∈S(X) a l l = p X 1 - a p p -1 = p X 1 - 1 p -1 p X p -1 p -a p .
For the first product, we use Lemma 4.2.1 to derive that

p X 1 - 1 p -1 e γ log X 1 - 1 2(log X) 2 .
For the second one, since

-log p X p -1 p -a p p X p (p -1)X 1 + O 1 log X 1 log X , we have p X p -1 p -a p 1 - 1 log X + O 1 (log X) 2 . It follows that I 2 (R; T ) I 1 (R; T ) e γ log X -1 + O 1 log X . ( 4
a 2 n = log p X 1 -a 2 p -1 = log p X 1 -(1 -p/X) 2 -1
In this subsection, we focus on the lower bounds of Γ σ (N ) defined in (4.1.9) and prove Theorem 4.1.2. The study of Γ σ (N ) arises naturally in metric Diophantine approximation. When σ = 1, this was a prize problem posed by the Dutch Mathematical Society in 1947 on Erdös's suggestion. Gál [START_REF] Gál | A theorem concerning Diophantine approximations[END_REF] investigated the problem in 1949 and proved that

Γ 1 (N ) (log 2 N ) 2 .
Thereafter, the GCD sums (4.1.8) are also known as "Gál-type sums". In 2017, Lewko and Radziw l l [START_REF] Lewko | Refinements of Gál's theorem and applications[END_REF] used the method of probabilistic models to give a much easier proof of Gál's theorem. They further determined the implied constant and proved that as N → ∞, one has

Γ 1 (N ) = e 2γ ζ(2) + o(1) (log 2 N ) 2 .
Let M be a finite set of integers. For general σ, define the spectral norm of the GCD matrix (gcd(m, n

) σ /lcm(m, n) σ ) (m,n)∈M 2 as Q σ (M) := sup c∈C |M| c 2 =1 m,n∈M c m c n gcd(m, n) σ lcm(m, n) σ ,
where c := (c 1 , . . . , c N ) ∈ C N and its norm c 2 := N j=1 |c j | 2 . Then in 2015, Aistleitner, Bondarenko and Seip [START_REF] Aistleitner | GCD sums from Poisson integrals and systems of dilated functions[END_REF] showed that

Γ 1 2 (N ) sup |M|=N Q 1 2 (M) (e 2 + 1)(log N + 2) max n N Γ 1 2 (n).
Recently, de la Bretèche and Tenenbaum [START_REF] De La Bretèche | Sommes de Gál et applications[END_REF] gave asymptotic formulas for Γ1 2 (N ). Namely, they proved that

Γ 1 2 (N ) = exp 2 √ 2 + o(1) log N log 3 N log N , (4.4.1)
as N → ∞. For further details about GCD sums, we refer to [START_REF] Aistleitner | GCD sums from Poisson integrals and systems of dilated functions[END_REF][START_REF] Bondarenko | Gál-type GCD sums beyond the critical line[END_REF][START_REF] Bondarenko | GCD sums and complete sets of square-free numbers[END_REF][START_REF] Lewko | Refinements of Gál's theorem and applications[END_REF]. Since Theorem 4.1.2 is a lower bound of Γ σ (N ), we only need to construct a set of integers M with |M| N such that

S σ (M) |M| exp 2 √ 2 + o(1) | log(σ -1 2 )| (log N ) 1-σ (log 2 N ) σ for 1 2 + 1 log 2 N
σ < 1 and N → ∞. Next we prove this by adapting the method of de la Bretèche and Tenenbaum [START_REF] De La Bretèche | Sommes de Gál et applications[END_REF].

Construction of the set M

Let α ∈ (1, +∞), η ∈ (0, +∞), f ∈ (1, e] and λ ∈ (0, 1) be some parameters. For 1 j J := (σ -1 2 ) -λ , define

I j := (f j (log N ) log 2 N, f j+1 (log N ) log 2 N ].
Therefore, by (4.4.2) and the definitions of v j and J, we have log |M|

S σ (M j ) = a,a |N j ω(a),ω(a ) v j gcd(a, a ) σ lcm(a, a ) σ b,b |N j gcd(b,a)=gcd(b ,a )=1 ω(b),ω(b ) v j gcd(b, b ) σ lcm(b, b ) σ . (4.4.7)
Denote the inner sum by S σ = S σ (a, a ), then we have

S σ = D,b,b |N j gcd(b,a)=gcd(b ,a )=1 gcd(b,b )=D ω(b),ω(b ) v j D 2σ (bb ) σ = d|N j gcd(d,aa )=1 ω(d) v j ϕ 2σ (d) b,b |N j gcd(b,a)=gcd(b ,a )=1 d| gcd(b,b ) ω(b),ω(b ) v j 1 (bb ) σ = d|N j gcd(d,aa )=1 ω(d) v j ϕ 2σ (d) d 2σ B,B |N j gcd(B,ad)=gcd(B ,a d)=1 ω(B),ω(B ) v j -ω(d) 1 (BB ) σ ,
where ϕ 2σ (d) is the Euler's totient function of order 2σ, satisfying d|D ϕ 2σ (d) = D 2σ . By the definition of ϕ 2σ (d), for d|N j we have

ϕ 2σ (d) d 2σ = p|d 1 - 1 p 2σ p|N j 1 - 1 p 2σ 1. (4.4.8)
Consequently, we derive that

S σ d|N j gcd(d,aa )=1 ω(d) v j B,B |N j gcd(B,ad)=gcd(B ,a d)=1 ω(B),ω(B ) v j -ω(d) 1 (BB ) σ .
Substitute this into (4.4.7). For the sum over a and a , we follow a similar procedure and derive that

S σ (M j ) c|N j ω(c) v j A,A |N j gcd(A,c)=gcd(A ,c)=1 ω(A),ω(A ) v j -ω(c) 1 (AA ) σ d|N j gcd(d,AA c)=1 ω(d) v j B,B |N j gcd(B,Acd)=gcd(B ,A cd)=1 ω(B),ω(B ) v j -ω(d) 1 (BB ) σ
Therefore, by (4.4.5) we have

c|N j ω(c) v j -u j d|N j gcd(d,c)=1 ω(d) v j -u j 1 |M j | 4 v j P j 2u j = |M j | 4 (1 + o(1)) √ α j(f -1) 1 2 f j/2 √ log J 4u j ,
Combined with (4.4.11), we obtain that

S σ (M j ) |M j | (1 + o(1)) e √ α(f 1-σ -1) η(1 -σ) √ f -1 4u j
.

By the definition of u j , we have

j J u j = 4η(log N ) 1-σ √ log J(log 2 N ) σ j J 1 jf j(σ-1 2 ) + O(J) = (1 + o(1)) 4η √ log J(log N ) 1-σ f J(σ-1 2 ) (log 2 N ) σ = (1 + o(1)) 4η λ| log(σ -1 2 )|(log N ) 1-σ f (σ-1 2 ) 1-λ (log 2 N ) σ .
Therefore, by taking product over j, we obtain

S σ (M) |M| exp (1 + o(1)) 4η λ| log(σ -1 2 )|(log N ) 1-σ f (σ-1 2 ) 1-λ (log 2 N ) σ log e √ α(f 1-σ -1) η(1 -σ) √ f -1 . Write H := 4η √ λ f (σ-1 2 ) 1-λ log e √ α(f 1-σ -1) η(1 -σ) √ f -1 .
Note that (4.4.6) implies we need to restrict 2α log f 1. To get large value of H, we set

f → 1 + , 2α log f → 1 -, η → √ 2 2 , λ → 1 -.
Then H can be sufficiently close to 2 √ 2 and we finally derive that

Γ σ (N ) exp (2 √ 2 + o(1)) | log(σ -1 2 )| (log N ) 1-σ (log 2 N ) σ .
4.5 Convolution method: Proof of Theorem 4.1.3

Let M be a set of positive integers with cardinal |M| = N = T κ where 0 < κ < 1 to be chosen. Define

M j = M 1 + log T T j , 1 + log T T j+1 .
For J = {j 0 : M j = ∅}, let M = {m j = min M j : j ∈ J }.

Then we define the resonator

R(t) = m∈M r(m)m it , where r(m j ) = |M j | 1
2 . Trivially we have

|R(t)| R(0) = m∈M r(m) m∈M 1 1 2 m∈M r(m) 2 1 2 |M | 1 2 |M| 1 2
N.

Let 0 < ε < 1, for u ∈ R, we take

K(u) := sin 2 (εu log T ) πu 2 ε log T , which satisfies K(ξ) = max 1 - |ξ| 2 log(T ε ) , 0 . (4.5.1) Write Z σ (t, u) := ζ(σ + it + iu)ζ(σ -it + iu)K(u).
Then we define

M 1 (R, T ) := T β |t| T |R(t)| 2 φ t log T T dt. M 2 (R, T ) := 2T β |t| T /2 |R(t)| 2 φ t log T T |u| |t|/2
Z σ (t, u) du dt.

Since K(•) is bounded by 1, clearly we have

Z β (σ, T ) 2 |M 2 (R, T )| M 1 (R, T ) . (4.5.2)
As in §3, we approximate M 1 (R, T ) and M 2 (R, T ) by their relative full integral, i.e., by

I 1 (R, T ) = R |R(t)| 2 φ t log T T dt, I 2 (R, T ) = R |R(t)| 2 φ t log T T R Z σ (t, u) du dt.
Lemma 4.5.1. For I (R, T ), we have

M 1 (R, T ) I (R, T ) T |M| log T ,
where the implied constant is absolute.

Proof. The first inequality is trivial. Further we have

I 1 (R, T ) = T log T i,j∈J r(m i )r(m j ) φ T log T log m j m i .
In the sum, the diagonal terms contribute

φ(0) i∈J r(m i ) 2 = φ(0) i∈J |M i | = φ(0)|M | φ(0)|M|. (4.5.3)
For the off-diagonal terms, we divide the sum according to the values of |i -j| and have

i,j∈J i =j r(m i )r(m j ) φ T log T log m j m i = l 1 i,j∈J |i-j|=l r(m i )r(m j ) φ T log T log m j m i l 1 i,j∈J |i-j|=l r(m i )r(m j ) φ T log T log 1 + log T T l-1
.

Recall that φ is rapidly decay. Thus this is bounded by

l 1 i,j∈J |i-j|=l r(m i )r(m j ) i∈J r(m i ) 2 |M|.
This combined with (4.5.3) proves the lemma.

Lemma 4.5.2. For I(R, T ), we have

I 2 (R, T ) = M 2 (R, T ) + O(|M|T β+κ log T ),
where the implied constant is absolute.

Proof. By the definition, we have that

I 2 (R, T ) -M 2 (R, T ) = |t|<2T β R + 2T β |t| T /2 |u|>|t|/2 + |t|>T /2 R |R(t)| 2 φ t log T T Z σ (t, u) du dt.
Denote the three integrals on the right-hand side by D 1 (R, T ), D 2 (R, T ) and D 3 (R, T ) respectively. We prove that each is bounded by O(|M|T β+κ log T ). Firstly, recall that

|ζ(σ + it)| (1 + |t|) 1 3 (1-σ)
. Therefore, we have

|t|<2T β |u| T β |Z σ (t, u)| du dt |t|<2T β |u| T β |ζ(σ + it)| 2 du dt T β log T, and
|t|<2T β |u|>T β |Z σ (t, u)| du dt |t|<2T β |u|>T β (|t| + |u|) 1 3 (1-σ) u 2 du dt T β .
Consequently, we derive that

D 1 (R, T ) N 2 |t|<2T β |u| T β + |u|>T β |Z σ (t, u)| du dt |M|T β+κ log T.
Secondly, we have that D 2 (R, T ) is bounded by

T β |t| T |R(t)| 2 φ t log T T |u|>|t|/2 (|t| + |u|) 1 3 (1-σ) u 2 du dt T -β/2 I (R, T ),
and thus is admissible by Lemma 4.5.1. Finally, since φ decays rapidly, we have

D 3 (R, T ) N 2 = |M|T κ .
This completes the proof.

Using Lemma 4.5.1 and Lemma 4.5.2, we establish from (4.5.2) that Then for any t = 0 we have

|M 2 (R, T )| M 1 (R, T ) I 2 (R, T ) T |M| log T + O T β+κ-1 (log T ) 2 . ( 4
R Z σ (t, u) du = k,l 1 K(log kl) k σ+it l σ-it -2πζ(1 -2it)K(i(σ + it) -i) -2πζ(1 + 2it)K(i(σ -it) -i).
We expand K(u) analytically continuously to the whole complex plane, satisfying the conditions of Lemma 4.5.3. Therefore, we may deduce that

I 2 (R, T ) = I 2,1 (R, T ) -I 2,2 (R, T ) -I 2,3 (R, T ), where I 2,1 (R, T ) = R |R(t)| 2 φ t log T T k,l 1 K(log kl) k σ+it l σ-it dt, I 2,2 (R, T ) = 2π R ζ(1 -2it)K(i(σ + it) -i)|R(t)| 2 φ t log T T dt, I 2,3 (R, T ) = 2π R ζ(1 + 2it)K(i(σ -it) -i)|R(t)| 2 φ t log T T dt.
For I 2,2 (R, T ) and I 2,3 (R, T ), we have

I 2,2 (R, T ) + I 2,3 (R, T ) |M|T κ log T R |ζ(1 ± 2it)| 1 + t 2 φ t log T T dt |M|T κ log T , (4.5.5)
since K(i(σ ± it) -i)

1/((1 + t 2 ) log T ). Thus I 2,2 (R, T ) and I 2,3 (R, T ) are admissible as error terms.

For I 2,1 (R, T ), each term is nonnegative, so we have

I 2,1 (R, T ) = T log T m,n∈M r(m)r(n) k,l 1 K(log kl) k σ l σ φ T log T log mk nl T log T m,n∈M r(m)r(n) kl T ε K(log kl) k σ l σ φ T log T log mk nl .
Note that by (4.5.1), kl T ε implies K(log kl) 1 2 . Therefore 4.6 Long resonance method: an alternative proof of (4.1.3)

In this section, we will use the long resonance method to detect large values of the Riemann zeta function in the strip 1 2 < e s < 1 without using the GCD sums. The following lemma states that for almost all t ∈ [0, T ], ζ(σ + it) can be approximated by its short Euler product. Lemma 4.6.1. Assume the zero-density estimates N (σ, T )

T A(σ)(1-σ)+o (1) . Then we have the short Euler product approximation of the Riemann zeta function for 1 2 < σ < 1 and 0 < λ < 1 fixed:

ζ(σ + it) = ζ(σ + it; Y ) 1 + O log |t|(log Y ) 2 Y λ(σ-1 2 )
, for all |t| ∈ [0, T ] except for a set E of measure at most Y T A(λ/2+(1-λ)σ)(1-λ/2-(1-λ)σ)+o (1) .

Proof. This follows directly from Lemma 3.2.3 by choosing σ 0 = λ/2 + (1 -λ)σ. S σ (t; Y )|R(t)| 2 φ t log T T dt + T A(λ/2+(1-λ)σ)(1-λ/2-(1-λ)σ)+2bσ+o (1) .

To extend the integral to the whole R, we need the following evaluations. By (4.6.1) we have We have the trivial lower bound for I 1 (R, T ) We have log p X

I 1 (R, T ) = R m,n∈S (
(1 -a 2 p ) -1 = 2σπ(X) log X -σθ(X)p X log(2X σ -p σ ). 3), we have I 1 (R, T ) T bσ(1+c(σ))+1+o (1) .

Thus E I 1 (R, T )
T A(λ/2+(1-λ)σ)(1-λ/2-(1-λ)σ)+bσ(1-c(σ))-1+o(1) + T β+bσ(1-c(σ))-1+o (1) .

I 2 (R, T ) I 1 (R, T ) p X a p p σ = p X 1 -(p/X) σ p σ = p X 1 p σ - π(X) X σ = (1 + o(1)) X σ (1 -σ) log X = b σ 1 -σ + o(1) (log T ) σ (log 2 T ) 1-σ ,
by Lemma 4.2.2 and the choice of X. Now we treat the inequality (4.6.4) and (4.6.5). Set β = 0, then we only need to do with (4.6.4). Since λ/2 + (1 -λ)σ can be as close to σ as possible, for any fixed small δ = δ(σ). Then (4. 5 Large values of |ζ ( ) (1 + it)|

Background

The study of the extreme values of the Riemann zeta function has a long history. Over the past decades, quite a few of results have been established. The values on the critical line σ = 1 2 were first considered by Titchmarsh [START_REF] Titchmarsh | The theory of the Riemann zeta-function, 2nd edn[END_REF], who showed that there exists arbitrarily large t such that for any α < 1 2 we have |ζ( 1 2 + it)| exp (log t) α . We refer to [START_REF] Montgomery | Extreme values of the Riemann zeta function[END_REF][START_REF] Balasubramanian | On the frequency of Titchmarsh's phenomenon for ζ(s), III[END_REF][START_REF] Balasubramanian | On the frequency of Titchmarsh's phenomenon for ζ(s)[END_REF][START_REF] Bondarenko | Large greatest common divisor sums and extreme values of the Riemann zeta function[END_REF][START_REF] Bondarenko | Note on the resonance method for the Riemann zeta function, 50 years with Hardy spaces[END_REF][START_REF] De La Bretèche | Sommes de Gál et applications[END_REF] for results in progression. For the critical strip 1 2 < e s < 1, it was also Titchmarsh [START_REF] Titchmarsh | On an inequality satisfied by the zeta-function of Riemann[END_REF] who first showed that for any ε > 0 and fixed σ ∈ ( 1 2 , 1), there exists arbitrarily large t such that |ζ(σ + it)| exp{(log t) 1-σ-ε }. We refer to [START_REF] Levinson | Ω-theorems for the Riemann zeta-function[END_REF][START_REF] Montgomery | Extreme values of the Riemann zeta function[END_REF][START_REF] Aistleitner | Lower bounds for the maximum of the Riemann zeta function along vertical lines[END_REF][START_REF] Bondarenko | Note on the resonance method for the Riemann zeta function, 50 years with Hardy spaces[END_REF][START_REF] Dong | On large values of |ζ(σ + it)[END_REF] on this topic. The study of the values on the 1-line can date back to 1925 when Littlewood [START_REF] Littlewood | On the Riemann zeta-function[END_REF] showed that there exists arbitrarily large t for which |ζ(1 + it)| {1 + o(1)}e γ log 2 t. Here and throughout, we denote by log j the j-th iterated logarithm and by γ the Euler constant. For further results, we refer to [START_REF] Levinson | Ω-theorems for the Riemann zeta-function[END_REF][START_REF] Granville | The Riemann Zeta Functionn and Related Themes: Papers in Honour of Professor K. Ramachandra[END_REF][START_REF] Aistleitner | Extreme values of the Riemann zeta function on the 1-line[END_REF][START_REF] Dong | On large values of |ζ(σ + it)[END_REF].

It also draws wide interests on the extreme values of the derivatives of the Riemann zeta function. For any ∈ N + , denote

Z ( ) (T ) := max t∈[T,2T ] |ζ ( ) (1 + it)|.
Besides other results, Yang [START_REF] Yang | Extreme values of derivatives of the Riemann zeta function[END_REF] recently proved that if T is sufficiently large, then uniformly for (log T )/(log 2 T ), we have In this section, we aim to improve the constant /( +1) +1 in (5.1.1). We have the following theorem. where c is a computable constant. Granville and Soundararajan [START_REF] Granville | The Riemann Zeta Functionn and Related Themes: Papers in Honour of Professor K. Ramachandra[END_REF] predicted that this is still true for max t∈[T,2T ] |ζ(1 + it)|. These results seems stronger than that of Theorem 5.1.1 with = 0. The reason is that after taking derivatives of the Riemann zeta function, we are no longer able to make use of the multiplicativity of its Dirichlet coefficients as was previously done. Nevertheless, Theorem 5.1.1 remains a generalization of Littlewood's initial bound (see [START_REF] Littlewood | On the Riemann zeta-function[END_REF]). Note that k ∈ M j \ M j-1 implies k x (j-1)/J . Therefore For each 0 i J, we rewrite uniquely m = m 1 m 2 where P + (m 1 ) x i/J and P -(m 2 ) > x i/J . Then we have

k∈M i 1 k m∈M k|m 1 = m 1 ∈M i k|m 1 1 k m 2 ∈M P -(m 2 )>x i/J 1.
For the sum over m 1 , we have

m 1 ∈M i k|m 1 1 k = p∈M i m 1 |p b-1 k|m 1 1 k = p x i/J b-1 ν=0 b -ν p ν .
For the sum over m 2 , clearly we have Therefore, we deduce that Note that b π(x) = |M|. By Lemma 5.2. While this is trivial by the integral inequalities

1 J J j=1 j -1 J 1 0 u du 1 J J j=1 j -1 J + 1 J .

Proof of Theorem 5.1.1

We start with the following lemma, which helps approximate the derivatives of the Riemann zeta function by the Dirichlet polynomials.

Lemma 5.3.1. For T → ∞, T t 2T and (log T )/(log 2 T ), we have that

(-1) ζ ( ) (1 + it) = n T (log n) n 1+it + O (log 2 T ) ,
where the implied constant is absolute.

Proof. This is [55, Lemma 1], where we have taken σ = 1 and ε = (log 2 T ) -1 as Yang did.

See also [START_REF] Titchmarsh | The theory of the Riemann zeta-function, 2nd edn[END_REF]Theorem 4.11].

To employ the resonance method, we choose the same weight function φ(•) as that used by Soundararajan [48, page 471]. Thus let φ(t) be a smooth function compactly supported in [START_REF] Aistleitner | Lower bounds for the maximum of the Riemann zeta function along vertical lines[END_REF][START_REF] Aistleitner | GCD sums from Poisson integrals and systems of dilated functions[END_REF], such that 0 φ(t) 1 always and φ(t) = 1 for t ∈ (5/4, 7/4). Then the Fourier transform of φ satisfies φ(u) α |u| -α for any integer α 1.

For sufficiently large T , we set x = log T 3 log 2 T and b = log 2 T .

Contents0 1 Preliminaries 1 . 1 1 . 5

 11115 Introduction en français 0.1 Sur la ligne critique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.2 Sur la ligne 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.3 Dans la bande 1 2 < e s < 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.4 Plan de cette thèse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 Introduction 0.1 On the critical line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.2 On the 1-line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.3 In the strip 1 2 < e s < 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.4 Outline of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Basic properties of the Riemann zeta function . . . . . . . . . . . . . . . . . 1.2 Zero-density estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Resonance method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 GCD sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Special case of square-free numbers . . . . . . . . . . . . . . . . . . . 1.4.3 Special case of friable numbers . . . . . . . . . . . . . . . . . . . . . . v Probabilistic models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Models for the Riemann zeta function . . . . . . . . . . . . . . . . . . 1.5.2 Models for the GCD sum . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.3 Models for L(1, χ d ) . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

  Large values of |ζ ( ) (1 + it)| 5.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Proof of Proposition 5.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Proof of Theorem 5.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography vii List of Symbols ζ(s) the Riemann zeta function log j x the j-th iterate of the natural logarithm of x χ the Dirichlet character ω(n) the number of distinct prime factors of n Ω(n) the number of all prime factors of n e the natural logarithm γ the Euler constant e s the real part of s m s the imaginary part of s C the set of complex numbers R the set of real numbers Z the set of integers N the set of natural numbers N + the set of positive integers E(X) the mathematic expectation of X gcd(m, n) the greatest common divisor of m and n lcm(m, n) the least common multiple of m and n 0 Introduction en français La fonction zêta de Riemann ζ(s) joue un rôle central dans la théorie analytique des nombres et a des applications en physique, en théorie des probabilités et en statistique appliquée.

β = 1 2 ,

 2 nous pouvons choisir la constante c = -3.6931472. Cela donne une description du terme d'erreur O(1) dans le résultat par Aistleitner, Mahatab et Munsch. Malgré l'intervalle élargi, le Théorème 0.2.2 est conforme à la Conjecture (0.2.3) qui prédit une constante plus grande C 0 + 1 -log 2 = -0.0885469.

  Nous visons à améliorer le facteur constant /( + 1) +1 dans (0.2.4). Nous avons le résultat suivant. Théorème 0.2.3. (Theorem 5.1.1) Nous avons Z ( ) (T ) e γ + 1 (log 2 T ) +1 {1 + o(1)}, uniformément pour T → ∞ et tous les entiers positifs (log T )/(log 2 T ).

  got a much stronger result max t∈[1,T ] |ζ(1 + it)| e γ {log 2 T + log 3 T -log 4 T + O(1)} holds for sufficiently large T . Then in 2019, Aistleitner, Mahatab, and Munsch [3] canceled the term log 4 T : max t∈[ √ T ,T ] |ζ(1 + it)| e γ {log 2 T + log 3 T + O(1)}. (2.1.1)

  generalize (2.1.4) to the joint distribution of arg ζ(1+it) and |ζ(1 + it)|.

Lemma 2 . 2 . 2 .

 222 Let y 2 and |t| y + 3 be real numbers. Let 1 2 σ 0 < 1 and suppose that the rectangle {z : σ 0 < e z 1, | m z -t| y + 2} is free of zeros of ζ(z). Then for any σ 0 < σ 2 and |ξ -t| y, we have | log ζ(σ + iξ)| log |t| log(e/(σ -σ 0 )).

Lemma 2 . 2 . 3 .

 223 Let {b(p)} p primes be a complex sequence. Then we have 1 T 2T T y p z b(p) p it 2k dt k y p z |b(p)| 2 k + 1 T 2/3 y p z |b(p)| 2k uniformly for T 8, 2 y z T

2 .

 2 Choosing k = [log T /(3 log z)] = [log T /(300 log 2 T )] and x(p) = 1/p, then by Lemma 2.

2 T

 2 )4 to (log T )(log 2 T ) A with arbitrary A > 0. Note that this change is harmless to the range of k, since whether y = (log T )(log 2 T )4 or y = (log T )(log 2 T ) A , it does not influence the upper bound of k: k log T e 10 log(y/ log T ) log T e 10 log(e 2 log T / log T ) = log T 2e 10 • (2.4.6)

2. 5 1 Firstly

 51 Proof of Theorem 2.1.we recall the definition of of the short Euler products ζ(s; y)

0 Φ 0 Φ 0 Φ 0 Φ T (t; y) dt 1 -a/b ∞ 0 Φ.

 000010 γ |ζ(1 + it; y)|) 2κ dt, i.e. 2κ ∞ T (t; y)t 2κ-1 dt = e -2κγ T 2T T |ζ(1 + it; y)| 2κ dt. (2.5.5) Now (2.5.3) follows from Propositions 2.3.1 and 2.4.1 when κ is an integer. Next let κ / ∈ N be a real number verifying (2.5.4). There is a unique integer k verifying (2.5.4) such that k -1 < κ < k. The formula (2.5.5) with κ = 1 2 and [26, Theorem 3] imply that ∞ T (u; y) du = e -γ T 2T T |ζ(1 + it; y)| dt e -γ 1 T b > a > 0, by the Hölder inequality, it follows that ∞ T (t; y)t a dt ∞ T (t; y)t b dt a/b Thus there are two absolute positive constants C and D such that

∞ 0 Φ 0 Φ 0 Φ 0 Φ

 0000 T (t; y)t a dt C ∞ T (t; y)t b dt a/b , ∞ T (t; y)t b dt D ∞ T (t; y)t a dt b/a .Applying the first inequality with (a, b) = (2κ -1, 2k -1) and the second inequality with (a, b) = (2k -3, 2κ13) respectively, we can obtain that

  studied the distribution of large values of |ζ(σ + it)| as t varies in [T, 2T ]. Let T be sufficiently large. Define the distribution function by Φ T (τ ) := 1 T meas t ∈ [T, 2T ] : log |ζ(σ + it)| > τ . (3.1.2)

Lemma 3 . 3 . 2 . 2 , 1 )

 33221 Let σ ∈ ( 1 be a fixed constant. Then we have1 T 2T T |ζ(σ + it; y)| k dt = n∈S(y) d k/2 (n) 2 n 2σ + O exp -log T 4 log y ,uniformly for (T, y, k) in (3.3.1), where the implied constant depends on σ only.Proof. This is a special case of Proposition 4.1 of[START_REF] Lamzouri | On the distribution of extreme values of Zeta and L-functions in the stip 1 2 < σ < 1[END_REF].Nowwe are ready to prove Proposition 3.3.1. Proof of Proposition 3.3.1. In view of Lemma 3.3.2, it is sufficient to show that n∈S(y)

.3. 7 )

 7 Now the required (3.3.3) follows from (3.3.4), (3.3.5), (3.3.6) and (3.3.7).

3 . 1 )

 31 , where C n is defined in (3.3.2) and the implied constant depends only on N and σ. + it; y)|) k dt, the required result of Lemma 3.4.2 follows from Proposition 3.3.1 immediately.

and of b * 0 = 1 .

 1 Thus Lemma 3.4.3 follows from recurrence.

max t∈[ 1

 1 ,T ] |ζ(1 + it)| e γ (log 2 T + log 3 T -log 4 T + O(1)) holds for sufficiently large T . Based on a distribution function which exhibits slightly smaller values, they also proposed a strong conjecture that max t∈[T,2T ] |ζ(1 + it)| = e γ (log 2 T + log 3 T + C 0 + 1 -log 2 + o(1)), (4.1.1)

  in 2009 separately. In 2018, Aistleitner, Mahatab and Munsch [3] used a variant "long resonance" to show that max t∈[ √ T ,T ] |ζ(1 + it)| e γ (log 2 T + log 3 T + O(1)).(4.1.2)

Theorem 4 . 1 . 1 . 4 - 1 . 1 2

 411411 Let 0 < β < 1 be fixed and c be a constant such that c < log(1 -β)log 2 Then we have Z β (1, T ) e γ (log 2 T + log 3 T + c) holds for sufficiently large T , When β = 1 2 , we can choose the constant c = -2.0197814. This gives a description of the error term O(1) in the result (4.1.2) by Aistleitner, Mahatab and Munsch. When β tends to 0, we can further choose c = -1.32663426. Despite the enlarged range, This is comparable with the conjecture (4.1.1) which predicts a larger constant C 0 + 1 -log 2 = -0.0885469. Theorem 4.1.1 is also in accordance with the results on the Dirichlet L-functions, due to Aistleitner, Mahatab, Munsch and Peyrot [4]. Now we turn our attention to the values of the Riemann zeta function in the strip < e s < 1.

S 2 (

 2 σ (M) := m,n∈M gcd(m, n) σ lcm(m, n) σ = m,n∈M gcd(m, n) 2σ (mn) σ . (4.1.8) For any positive integer N , we denote Γ σ (N ) := sup |M|=N S σ (M) N . (4.1.9)A brief historic description on Gál-type sums will be presented in §4.4. By adapting the argument of de la Bretèche and Tenenbaum[START_REF] De La Bretèche | Sommes de Gál et applications[END_REF] about Γ 1 N ), we have the following theorem concerning the lower bound of Γ σ (N ).

Theorem 4 . 1 . 2 .σ 1 2

 4121 As , we have

Corollary 4 . 1 . 4 .+ 1 log 2 T < σ < 3 4 .

 41424 Let T → ∞ and 1 2 Then we have that max t∈[T β ,T ]

Lemma 4 . 2 . 3 . 2 1 . 1 - 1 p w - 1 = p ν 1 1 νp νw = n 2 Λ= σ 1 +iT 0 σ 1 -iT 0 + σ 0 +iT 0 σ 1 +iT 0 + σ 0 -iT 0 σ 0 +iT 0 + σ 1 .

 423211112010001 Let a, a , b, b ∈ N + such that gcd(a, b) = 1 and gcd(a , b ) = 1. Then for any N ∈ N + satisfying b|N and b |N , we have a ) gcd(b, b ) . As a direct inference, we have gcd(N a/b, N a /b ) lcm(N a/b, N a /b ) = gcd(a, a ) lcm(a, a ) gcd(b, b ) lcm(b, b ) . Proof. Since lcm(b, b )|N , we may write gcd N a b , N a b = N gcd(a, a ) lcm(b, b ) gcd lcm(b, b ) b a gcd(a, a ) , lcm(b, b ) b a gcd(a, a ) = N gcd(a, a ) lcm(b, b ) gcd b gcd(b, b ) a gcd(a, a ) , b gcd(b, b ) a gcd(a, a ) . Then the first assertion follows by the assumptions of co-primeness and the second one follows by the simple fact that gcd gcd(a, a )lcm(a, a ) gcd(b, b )lcm(a, a ) . For convenience, we denote the truncated Euler product of the Riemann zeta function by ζ(s; y) := The following lemma approximates ζ(s) on the 1-line by its truncated Euler product. Lemma 4.2.4. Let T be a large positive number and 0 < β < 1 be fixed. Put Y = exp((log T ) 1/β ). Then we have ζ(1 + it) = ζ(1 + it; Y ) 1 + O 1 (log T ) 1/β for any T β |t| T . Proof. Let s = 1 + it with T β |t| T . Firstly, when e w > 1, we have log ζ(w) = log p (n) n w log n , where Λ(n) is the von Mangoldt function. Set σ 1 = 1/ log Y , σ 0 = -c/ log T for suitably positive constant c = c(β) > 0 and T 0 = T β /2. Denote the contour joining σ 1 -iT 0 , σ 1 + iT 0 , σ 0 + iT 0 , σ 0 -iT 0 and σ 0 + iT 0 by Γ, i.e., Γ -iT 0 σ 0 -iT 0 Since log ζ(1 + it + w)Y w is analytic inside Γ, by Cauchy's integral formula we have 1 2πi Γ log ζ(1 + it + w) Y w w dw = log ζ(1 + it). (4.2.1) Then by Perron's formula ([49, Corollary II.2.1] with s = 1 + it, σ a = α = 1, κ = 1/ log Y and B(x) = 1), we have 1 2πi

4. 3

 3 Extreme values of |ζ(1+it)|: Proof of Theorem 4.1.1

  where θ(x) is the Chebyshev function. It is well known that π(x) log x -θ(x) = {1 + o(1)} x log x and thus we have |R(t)| T B+o(1) (t ∈ R). (4.3.1)

.5. 4 )Lemma 4 . 5 . 3 .

 4453 To estimate I(R, T ), we need to deal with the convolution of K(u) with ζ(s). Here we quote the following lemma due to de la Bretèche and Tenenbaum[START_REF] De La Bretèche | Sommes de Gál et applications[END_REF] Lemma 5.3]. Let σ ∈ (-∞, 1). Suppose K(z) is analytic in the strip m z ∈ [σ -2,

I 2 , 1 ( 6 ). 1 T 2 = 1 T (σ-1 2 )ε S 1 2 ( 2 T

 216121222 For the inner sum, for fixed k and l, we have m,n∈M r(m)r(n) φ T log T log mk nl i,j∈J m∈M i ,n∈M j mk=nl /m i 1 + log T /T and 1 n/m j 1 + log T /T . Thus we have I 2,1 (R, T ) Write m = m gcd(m, n) and n = n gcd(m, n). Then the relation mk = nl impliesL = k n = l m ,for some integer L andkl = L 2 m n = L 2 lcm(m, n) gcd(m, n) .Therefore in (4.5.6), we haveI 2,1 (R, T ) T log T m,n∈M L 2 lcm(m,n) gcd(m,n) T ε By Rankin's trick, we have m,n∈M lcm(m,n) gcd(m,n) >T ε gcd(m, n) lcm(m, n) σ (σ-1 2 )ε m,n∈M gcd(m, n) lcm(m, n)1 and we can choose ε = 1 2022 . So we haveI 2,1 (R, T ) T log T S σ (M).Combining this with (4.5.4) and (4.5.5), we have|M 2 (R, T )| M 1 (R, T ) S σ (M) |M| + T β+κ-1 (log T ) 2 .Choose κ = 1 -β, by (4.5.2) we haveZ β (σ, T ) 2 |M 2 (R, T )| M 1 (R, T ) S σ (M) |M| + (log T ) 2 ,and thus by taking maximum of both sides over |M| = N we haveZ β (σ, T ) Γ σ (T 1-β ).

For X = b log T log 2 T

 2 with b = b(σ) positive to be determined later, we define the resonator R(t) := p X(1 -a p p it ) -1 = n∈S(X)a n n it ,wherea p = 1 -p X σfor p X, and a p = 0 for p > X,and we extend a n such that it is completely multiplicative. Since logR(0) = log p X X p σ = σ(π(X) log X -θ(X)) = σ(1 + o(1)) X log X ,we have|R(t)| R(0) T bσ+o(1) . (4.6.1)Choosing Y = (log T ) 3/(λ(σ-1 2 )) , then we haveζ(σ + it) = ζ(σ + it; Y ) 1 + O 1 log T , for all t ∈ [-T, T ] \ E with meas E T A(λ/2+(1-λ)σ)(1-λ/2-(1-λ)σ)+o(1) . (4.6.2) Since ζ(σ + it; Y ) = Clearly we have S σ (t; Y ) T o(1)by the choice of Y . The relationZ β (σ, T ) := max T β |t| T |ζ(σ + it)| max |t|∈[T β ,T ]\E |ζ(σ + it)| max |t|∈[T β ,T ]\E |ζ(σ + it; Y )| exp max |t|∈[T β ,T ]\E e S σ (t; Y )shows that it suffices to determine the maximum inside the exponent. DefineM 2 (R, T ) := |t|∈[T β ,T ]\E S σ (t; Y )|R(t)| 2 φ t log T T dt.Then by (4.6.1) and (4.6.2) we haveM 2 (R, T ) = |t|∈[T β ,T ] S σ (t; Y )|R(t)| 2 φ t log T T dt -|t|∈E S σ (t; Y )|R(t)| 2 φ t log T T dt = |t|∈[T β ,T ] S σ (t; Y )|R(t)| 2 φ t log T T dt + O((meas E)R(0) 2 )T o(1) = |t|∈[T β ,T ]

M 2

 2 |t|<T β S σ (t; Y )|R(t)| 2 φ t log T T dt R(0) 2 T β+o(1) T β+2bσ+o(1) .The rapid decay of φ(•)implies |t|>T S σ (t; Y )|R(t)| (R, T ) = R S σ (t; Y )|R(t)| 2 φ t log T T dt + O(T A(λ/2+(1-λ)σ)(1-λ/2-(1-λ)σ)+2bσ+o(1) + T β+2bσ+o(1) ) :=I 2 (R, T ) + O(T A(λ/2+(1-λ)σ)(1-λ/2-(1-λ)σ)+2bσ+o(1) + T β+2bσ+o(1) ) =I 2 (R, T ) + E,where we have denoted the sum of the two error terms by E = E(σ, λ, β, a, b). Note thatI 2 (R, T ) ∈ R + . Now since M 1 (R, T ) := |t|∈[T β ,T ]\E |R(t)| 2 φ t log T T dt R |R(t)| 2 φ t log T T dt := I 1 (R, T ), we have max |t|∈[T β ,T ]\E e S σ (t; Y ) e M 2 (R, T ) M 1 (R, T ) = e I 2 (R, T ) + E M 1 (R, T ) I 2 (R, T ) I 1 (R, T ) + E I 1 (R, T ).

( 1

 1 Sincep X log(2X σ -p σ ) = σπ(X) log X + σ -a 2 p ) -1 = σ{1 + c(σ) + o(1)} X log X = bσ{1 + c(σ) + o(1)} log T,and then by (4.6.

2 (

 2 AssumeA(λ/2 + (1 -λ)σ)(1 -λ/2 -(1 -λ)σ) + bσ(1 -c(σ)) -1 < 0 (4.6.4)andβ + bσ(1 -c(σ)) -1 < 0. (4.6.5)Then we only need to bound the ratio I 2 (R, T )/I 1 (R, T ). Since I

  6.5) implies b < 1 -A(σ)(1 -σ) σ(1 -c(σ)) -δ/2.

Z

  ( ) (T ) e γ ( + 1) +1 {log 2 T -log 3 T + O(1)} +1 .(5.1.1)

Theorem 5 . 1 . 1 .

 511 For T → ∞ and (log T )/(log 2 T ), we haveZ ( ) (T ) e γ + 1 (log 2 T ) +1 {1 + o(1)}.Remark 1. Recently, we[START_REF] Dong | On large values of |ζ(σ + it)[END_REF] proved that maxt∈[ √ T ,T ]|ζ(1 + it)| e γ (log 2 T + log 3 T + c),

1 (

 1 Now we are prepared to prove Proposition 5.1.1. By the construction, the set M is divisor-closed which means k | m, m ∈ M implies k ∈ M. Then by the definition of M j in (5.1.3), we have m∈M k|m (log k) k = J i=1 k∈M j \M j-

  Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Preliminary lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Preliminary lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A brief review on Gál-type sums . . . . . . . . . . . . . . . . . . . . . 4.4.2 Construction of the set M . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Proof of Theorem 4.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . .

	2 Distribution of |ζ(1 + it)| 4.4 Lower bound for Γ σ (N ) as σ 2.1 4.3 Extreme values of |ζ(1 + it)|: Proof of Theorem 4.1.1 . . . . . . . . . . . . . 1 2 : Proof of Theorem 4.1.2 . . . . . . . . . . 4.4.1

2.3 An asymptotic development . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Moments of the short Euler products . . . . . . . . . . . . . . . . . . . . . . 2.4.1 A preliminary lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Proof of Proposition 2.4.1 . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Proof of Theorem 2.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Two preliminary lemmas . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Proof of Proposition 2.5.1 . . . . . . . . . . . . . . . . . . . . . . . . 2.5.3 End of the proof of Theorem 2.1.1 . . . . . . . . . . . . . . . . . . . . 3 Distribution of |ζ(σ + it)| in the strip 1 2 < σ < 1 3.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Preliminary lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Moments of the short Euler products . . . . . . . . . . . . . . . . . . . . . . 3.4 Proof of Theorem 3.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Two preliminary lemmas . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Proof of Proposition 3.4.1 . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 End of the proof of Theorem 3.1.1 . . . . . . . . . . . . . . . . . . . . vi 4 Large values of |ζ(σ + it)| 4.1 4.5 Convolution method: Proof of Theorem 4.1.3 . . . . . . . . . . . . . . . . . . 4.6 Long resonance method: an alternative proof of (4.1.3) . . . . . . . . . . . .

  1-σ /(log t) σ . Dans les trois sections suivantes, nous présenterons ces sujets en détail.

	0.1 Sur la ligne critique
	Titchmarsh [51, Théorème 8.12] a montré qu'il existe un t arbitrairement grand tel que pour
	tout α < 1 2 ,
	|ζ( 1 2 + it)| exp((log t) α ).
	En 1977, sous l'hypothèse de Riemann (RH), Montgomery [41] a obtenu une meilleure borne
	inférieure
	|ζ( 1 2 + it)| > exp 1 20	log t log 2 t
	valable pour un t arbitrairement grand, où log j désigne le j-ème logarithme itéré. En 1977,
	Balasubramanian et Ramachandra [7] ont montré inconditionnellement (sans RH) qu'il existe
	un t arbitrairement grand tel que
	|ζ( 1

2 + it)| > exp c log t log 2 t , où c > 0 est une constant. En 1986, Balasubramanian

[START_REF] Balasubramanian | On the frequency of Titchmarsh's phenomenon for ζ(s)[END_REF]

a réussi à donner une valeur effective pour cette constante : c = 0, 530 . . . . En 2008, Soundararajan

[START_REF] Soundararajan | Extreme values of zeta and L-functions[END_REF]

a montré par la méthode de résonance que c = 1 + o(1). En 2017, Bondarenko et Seip

[START_REF] Bondarenko | Large greatest common divisor sums and extreme values of the Riemann zeta function[END_REF]

ont fait un progrès significatif en établissant une borne inférieure de type max t∈[0,T ]

  This is [26, Lemma 3]. The following lemma is an approximation of ζ(s) by ζ(s; y). Lemma 2.2.4. Let T 2 and y log T . Then we have Proof. This is essentially [26, Proposition 1] while we erase restrictions of the upper bound of y. In fact, the truncated Euler product with larger length would provide a better approximation of the zeta function. Without loss of generality, we assume y (log T ) 100 . Firstly we use Lemma 2.2.2 to approximate ζ(1 + it) by ζ(1 + it; z) with z relatively large. Choosing z = (log T ) 100 , then by Lemma 2.2.2 we have

	ζ(1 + it) = ζ(1 + it; y) 1 + O	(log T )/y log 2 T
	for all t ∈ [T, 2T ] except for a set of measure at most O(T exp{-(log T )/(50 log 2 T )}).

1/3

and all integers 1 k log T /(3 log z), where the implied constant is absolute.

Proof.

ζ(1 + it) = ζ(1 + it; z)(1 + O(1/ log T ))

for all t ∈ [T, 2T ] but at most a set of measure of T 4/5 . Then we use Lemma 2.2.3 to approximate ζ(1 + it; z) by ζ(1 + it; y) since ζ(1 + it; z) = ζ(1 + it; y) exp

  .3.7) Now Proposition 2.3.1 follows from (2.3.3), (2.3.6) and (2.3.7) with C j = C * j + C * * j . 2.4 Moments of the short Euler products Let ζ(s; y) and D k (y) be defined as in (2.1.8) and (2.3.1), respectively. In this section, we shall evaluate the 2k-th moment of ζ(1 + it; y) by proving the following proposition. This is essentially the first part of Theorem 3 in [26]. The main difference is a slightly enlarged length of the short Euler products, which is important for the proof of Theorem 2.1.1.

  in form

					k<p y	log p p	log	25y k	,
	we can derive that
				S k (I 0 )	D k,0 G δ 0	exp	3ky δ log k	log	25y k	.
	Taking δ = 1/(10 log 2 T ) and noticing that t → (t/ log t) log(25y/t) is increasing in I 0 , we
	deduce, for (T, y, k) in (2.4.1),
	ky δ log k	log	25y k	e 1/5 (log T )/(e 10 log(y/ log T )) log((log T )/(e 10 log(y/ log T )))	log	25e 10 y log T	log	y log T
					10e 1/5 log T
				e 10 log((log T )/(e 10 log(y/ log T )))
				20e 1/5 log T e 10 log 2 T	•
	Inserting this into the preceding inequality, we have

  into two parts

	+	(-1) |W|	(2.4.17)
	mr,nr∈S(Ir) gr Gr	W⊂R W =∅	mr,nr∈S(Ir), ∀ r∈R gw>Gw,∀w∈W
	∀r∈R

  log 2 T )4

			(2.4.24)
	since 2 L+1	exp(2 log 2 k)	exp(2 log 3 T ). Now the desired result follows from (2.4.18) and
	(2.4.24).

  2.5.3 End of the proof of Theorem 2.1.1

	By Lemma 2.2.4, we can derive that
	Φ T (τ ) = Φ T (τ + O(ε + η); y) + O(exp(-(log T )/(50 log 2 T ))	(2.5.23)
	with η := (log T )/y. Combining (2.5.22) and (2.5.23), we can obtain

  2v j {j log f + O(log j + log 4 N )} 2α log f log N + o(log N ),Note that ab|N j implies gcd(a, b) = 1 since N j is square-free. By Lemma 4.2.3, we have

	1 j J
	and consequently we obtain that
	|M| N 2α log f +o(1) .	(4.4.6)
	4.4.3 Proof of Theorem 4.1.2

  1, we haveIn view of (5.2.2), by taking difference of M j-1 and M j , we obtain k∈M j \M j-1

		k∈M i	1 k	k|m m∈M	1 =	i J	|M| 1 + O	log 2 x b	+	J log x	e γ log x.	(5.2.2)
				1 k	m∈M	1 =	|M| J	1 + O	J log 2 x b	+	J 2 log x	e γ log x.
					k|m
	Inserting this into (5.2.1), we have
	m∈M k|m	(log k) k		|M| J	J j=1	j -1 J	1 + O	J log 2 x b	+	J 2 log x	e γ (log x) +1 .
	Now Proposition 5.1.1 follows supplied that
					1 J	J j=1		j -1 J	=	1 + 1	+ O	1 J	.

Acknowledgments

Proof. This is equation (2.1) of [START_REF] Lamzouri | On the distribution of extreme values of Zeta and L-functions in the stip 1 2 < σ < 1[END_REF]. See also [START_REF] Bondarenko | Note on the resonance method for the Riemann zeta function, 50 years with Hardy spaces[END_REF]Lemma 6], [START_REF] Norton | Upper bounds for sums of powers of divisor functions[END_REF]Lemma 3.1], and [START_REF] Bober | The distribution of the maximum of character sums[END_REF]Lemma 3.3].

We need to approximate Riemann zeta function ζ(s) by its short Euler product. The following lemma shows that when ζ(s) has no zero in a good region, it can be approximated well by its short Euler product. Lemma 3.2.3. Let σ 0 ∈ [ 1 2 , 1) be a fixed number. Let y 2 and |t| y + 3 be real numbers and suppose that the rectangle {z : σ 0 < e z 1 and | m z -t| y + 2} is free of zeros of ζ(z). Then for any σ 0 < σ 2 and |ξ -t| y, we have

(log |t|) log(e/(σ -σ 0 )).

Further for σ 0 < σ 1, we have

where σ 1 := min σ 0 + (log y) -1 , 1 2 (σ + σ 0 ) . The implied constants depend on σ 0 at most.

Proof. See [26, Lemma 1].

With the help of Lemma 3.2.3, as well as a result of zero density estimate for the Riemann zeta-function ζ(s), we can approximate ζ(s) by its short Euler product mostly often. Of course, here the short Euler product is a bit "long", that means y needs to be relatively large. Otherwise, the error term will be too large to make sense. Lemma 3.2.4. Let σ ∈ ( 1 2 , 1) be a fixed number and 0 < a(σ) < 1 2 (σ-1 2 ) < 2/(σ-1 2 ) < A(σ).

Then for

T → ∞ and (log T ) A(σ) y T a(σ)

the asymptotic formula

2 ) y(log T ) 5 ), where the implied constants depend on σ at most. Proof. This is essentially [26, Lemma 2] while we restrict (log T ) A(σ) y T a(σ) such that both the error term O(y -1 2 (σ-1 2 ) (log y) 2 log T ) and the measure T 1-1 2 (σ-1 2 ) y(log T ) 5 make sense. We replace the term O(y

The proof has no difference from that of [START_REF] Granville | The Riemann Zeta Functionn and Related Themes: Papers in Honour of Professor K. Ramachandra[END_REF]Lemma 2].

In order to approximate ζ(s) by its "shorter" Euler product, we need the following moment evaluation for the sum over complex power of primes between two large numbers y and z, where y can be relatively smaller.

we can write

Using this and Lemma 3.2.4 with y = z = (log T ) 6/(σ-1 2 ) , we obtain

i.e.

for all t ∈ [T, 2T ] but at most a set of measure of

By partial summation formula, we can derive that

For the first integral, we have

For the second one, the interval guarantees that

Thus we have

Combining with (4.3.6) and (4.3.7), we obtain that log n∈S(X)

which implies by (4.3.4) that (1) .

In view of (4.3.3), for the error term we have (1) , where the last exponent is negative and thus admissible. Substituting this and (4.3.5) into (4.3.3), we deduce that

where c< log B -1. This completes the proof.

A brief review on Gál-type sums

Then for each interval I j , we have that

where we have supposed that f 1 + ((log N ) log 2 N ) -5/12+ε such that the prime number theorem in short intervals holds and the implied constant depends on ε only. On the other hand we note that the hypothesis σ 1 2 + 1/ log 2 N guaranties J (log 2 N ) λ = o(log 2 N ). Let N j = p∈I j p and ω(•) counts the number of different prime factors. Set

Then we define

Now we evaluate the cardinal of M. For 1 j J, we have

For fixed k, since P j is much larger than v j , for 0 v j we can deduce that

Therefore, we have

On the other hand, we have

Thus, by (4.4.3) and (4.4.4), we deduce that

Using the Euler-Maclaurin formula, we have

We calculate from inside successively. Since each term is positive, we can sift a suitable subset. Therefore we restrict c, d such that ω(c) = ω(d) = v j -u j , and A, A such that ω(A) = ω(A ) = u j . Then the inner sum turns to

For the factorial, we use Stirling's formula

For the sums, by Lemma 4.2.2 we have

Note that J → ∞ as σ 1 2 . Therefore, in (4.4.10) we have

We can play similar trick on sums over B, A , A in (4.4.9) successively and therefore

(4.4.11)

Trivially, we have

We use Ingham's bound A(σ) 3/(2 -σ), then we have

So at last, we have obtained max

where b is as above. Both Yang's proof and ours employ the resonance method used by Bondarenko and Seip [START_REF] Bondarenko | Note on the resonance method for the Riemann zeta function, 50 years with Hardy spaces[END_REF]. For a large x, we take P := The key ingredient of the proof is a weighted reciprocal sum in the form

where 0 is an integer. In [START_REF] Yang | Extreme values of derivatives of the Riemann zeta function[END_REF], Yang divided M as well as P into two subsets, according to whether p x /( +1) . Our choice is to give a finer division. Specifically, let J 1 be a positive integer. For 0 j J, denote

(5.1.3) Thus we divide the set M into J subsets:

.

By this trick, we are able to enlarge the estimate of S(x; ) with a factor (1 + 1/ ) . We summarize it as the following proposition.

Proposition 5.1.1. Under the previous notation, we have

uniformly for x 3, b 1, J 1, 0 where the implied constant is absolute.

Proof of Proposition 5.1.1

The following asymptotic formula plays a key role in the proof of Proposition 5.1.1.

Lemma 5.2.1. We have

uniformly for x 3 and b 1, where the implied constants are absolute.

Proof. See also [START_REF] Yang | Extreme values of derivatives of the Riemann zeta function[END_REF]Eq. (15)] and [14, page 129]. For a fixed prime p, we have

Furthermore, we take P and M as (5.1.2). Note that P √ T by the prime number theorem. Then we define the resonator

Since supp(φ) ⊂ [1, 2], we have that

For M 1 (R, T ), we have