# Analytic theory of L-functions: explicit formulae, gaps between zeros and generative computational methods

Kühn, Patrick. Analytic theory of L-functions: explicit formulae, gaps between zeros and generative computational methods. 2016, University of Zurich, Faculty of Science.

## Abstract

Despite its unifying content, the theme of the thesis is very modular. The chapters are indeed fairly independent from each other. The central topic behind them are the L-functions, their arithmetic and analytic properties, related computational methods, and their connections to various mathematical objects.
Each chapter contains a detailed introduction and motivation to study its associated topic. We summarize each chapter below.

I. An introduction to the central objects of the thesis, namely the L-functions. The Selberg class is defined, the most important properties are explained and the approximate functional equation is shown.

II. A method to recover Dirichlet coefficients of self-dual L-functions is introduced. Moreover, the python implementation is explained and bounds for the relative error of the computed solution are computed.

III. The upper bound on the largest gap between consecutive zeros of general entire L-functions is improved from 45:3236 to 41:54 under GRH and the Ramanujan hypothesis. Moreover, a new conjecture about the lowest upper bound is stated. This chapter is taken from [KRZ].

IV. Properties of a new arithmetic function generalizing the Ramanujan sum are derived, Moreover, alternative Riemann hypothesis equivalences concerning this new arithmetic function are proved, and additional results about the Bartz functionare obtained. This is based on [KR16].

V. A class of functions that satisfies intriguing explicit formulae of Ramanujan andTitchmarsh involving the zeros of an L-function in the Selberg class of degreeone and its associated Möbius function is studied. Moreover, some applicationswith certain explicit examples are obtained. This material is from the first half of [KRR14].

VI. The mollification put forward by Feng is computed by analytic methods and thesituation of the percentage of the critical zeros of the Riemann-zeta function onthe critical line is explained. This chapter is based on the preprint [KRZ16]. Additional work during this degree but not covered in this thesis can be found in [KR16; KRR14; KRZ16; KRZ].

## Abstract

Despite its unifying content, the theme of the thesis is very modular. The chapters are indeed fairly independent from each other. The central topic behind them are the L-functions, their arithmetic and analytic properties, related computational methods, and their connections to various mathematical objects.
Each chapter contains a detailed introduction and motivation to study its associated topic. We summarize each chapter below.

I. An introduction to the central objects of the thesis, namely the L-functions. The Selberg class is defined, the most important properties are explained and the approximate functional equation is shown.

II. A method to recover Dirichlet coefficients of self-dual L-functions is introduced. Moreover, the python implementation is explained and bounds for the relative error of the computed solution are computed.

III. The upper bound on the largest gap between consecutive zeros of general entire L-functions is improved from 45:3236 to 41:54 under GRH and the Ramanujan hypothesis. Moreover, a new conjecture about the lowest upper bound is stated. This chapter is taken from [KRZ].

IV. Properties of a new arithmetic function generalizing the Ramanujan sum are derived, Moreover, alternative Riemann hypothesis equivalences concerning this new arithmetic function are proved, and additional results about the Bartz functionare obtained. This is based on [KR16].

V. A class of functions that satisfies intriguing explicit formulae of Ramanujan andTitchmarsh involving the zeros of an L-function in the Selberg class of degreeone and its associated Möbius function is studied. Moreover, some applicationswith certain explicit examples are obtained. This material is from the first half of [KRR14].

VI. The mollification put forward by Feng is computed by analytic methods and thesituation of the percentage of the critical zeros of the Riemann-zeta function onthe critical line is explained. This chapter is based on the preprint [KRZ16]. Additional work during this degree but not covered in this thesis can be found in [KR16; KRR14; KRZ16; KRZ].