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1	Integral Moments of Quadratic Dirichlet L-functions: A Computational Perspective Alderson, Matthew 27 April 2010 (has links) In recent years, the moments of L-functions has been a topic of growing interest in the field of analytic number theory. New techniques, including applications of Random Matrix Theory and multiple Dirichlet series, have lead to many well-posed theorems and conjectures for the moments of various L-functions. In this thesis, we theoretically and numerically examine the integral moments of quadratic Dirichlet $L$-functions. In particular, we exhibit and discuss the conjectures for the moments which result from the applications of Random Matrix Theory, number theoretic heuristics, and the theory of multiple Dirichlet series. In the case of the cubic moment, we further numerically investigate the possible existence of additional lower order main terms. integral moments quadratic Dirichlet L-functions quadratic Dirichlet characters Dedekind zeta function Pure Mathematics
2	Integral Moments of Quadratic Dirichlet L-functions: A Computational Perspective Alderson, Matthew 27 April 2010 (has links) In recent years, the moments of L-functions has been a topic of growing interest in the field of analytic number theory. New techniques, including applications of Random Matrix Theory and multiple Dirichlet series, have lead to many well-posed theorems and conjectures for the moments of various L-functions. In this thesis, we theoretically and numerically examine the integral moments of quadratic Dirichlet $L$-functions. In particular, we exhibit and discuss the conjectures for the moments which result from the applications of Random Matrix Theory, number theoretic heuristics, and the theory of multiple Dirichlet series. In the case of the cubic moment, we further numerically investigate the possible existence of additional lower order main terms. integral moments quadratic Dirichlet L-functions quadratic Dirichlet characters Dedekind zeta function Pure Mathematics
3	Statistique des zéros non-triviaux de fonctions L de formes modulaires / Statistics on non-trivial zeros of modular L-functions Bernard, Damien 09 December 2013 (has links) Cette thèse se propose d’obtenir des résultats statistiques sur les zéros non-triviaux de fonctions L. Dans le cas des fonctions L de formes modulaires, on prouve qu’une proportion positive explicite de zéros non-triviaux se situe sur la droite critique. Afin d’arriver à ce résultat, il nous faut préalablement étendre un théorème sur les problèmes de convolution avec décalage additif en moyenne de manière à déterminer le comportement asymptotique du second moment intégral ramolli d’une fonction L de forme modulaire au voisinage de la droite critique. Une autre partie de cette thèse, indépendante de la précédente, est consacrée à l'étude du plus petit zéro non-trivial d’une famille de fonctions L. Ces résultats sont en particulier appliqués aux fonctions L de puissance symétrique. / The purpose of this dissertation is to get some statistical results related to nontrivial zeros of L-functions. In the modular case, we prove and determine an explicit positive proportion of non-trivial zeros lying on the critical line. In order to obtain this result, we need to extend a theorem on shifted convolution sums on average to be able to determine the asymptotic behaviour of the mollified second integral moment of a modular L-function close to the critical line. Independently of these results, we study the smallest non-trivial zero in a family of L-functions. These results are applied to symmetric power L-functions. Fonction L Forme modulaire Zéros non-triviaux Proportion Levinson Problème de convolution Décalage additif Moments intégraux Second moment ramolli Théorème de densité Plus petit zéro Équation différentielle avec retards L-function Modular form Non-trivial zeros Proportion Levinson Shifted convolution sums Integral moments Mollified

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