On some problems involving Dirichlet L-functions
Smith, K
Date: 9 May 2023
Thesis or dissertation
Publisher
University of Exeter
Degree Title
Doctor of Philosophy
Abstract
In this thesis we study three problems in analytic number theory. These problems are related at a fundamental level, but a general introduction to those relationships would digress from the subsequent content and a summary would certainly be superficial. As such, each chapter includes a self-contained introduction and statement of ...
In this thesis we study three problems in analytic number theory. These problems are related at a fundamental level, but a general introduction to those relationships would digress from the subsequent content and a summary would certainly be superficial. As such, each chapter includes a self-contained introduction and statement of results so that it may be read independently, but references to more unified literature are given where appropriate.
Chapter \ref{ch3} addresses the problem of computing asymptotic formulae for the expected values and second moments of central values of primitive Dirichlet $L$-functions $L(s,\chi_{8d}\otimes\psi)$ when $\psi$ is a fixed even primitive non-quadratic character of odd modulus $q$, $\chi_{8d}$ is a primitive quadratic character and $d\equiv h\pmod r$ is odd and squarefree, $h$ is fixed and $r\equiv 0\pmod q$ is even. Restricting to these arithmetic progressions ensures that such sets of $L$-functions form a ``family of primitive $L$-functions'' in the specific sense defined by Conrey, Farmer, Keating, Rubinstein and Snaith. Soundararajan had previously computed these statistics without restricting to such arithmetic progressions. We find that this restriction introduces non-negligible non-diagonal terms to the second moments that require significantly more detailed analysis to handle.
Chapter \ref{chaplind} focuses on the moments and subconvexity of the Riemann zeta function $\zeta(\sigma+it)$ in the right-half of the critical strip $1/2<\sigma<1$ from a functional-analytic perspective. We examine a correspondence between the moments and the Hilbert space $B^2$ of Besicovitch almost-periodic functions and a certain subgroup $U$ of its unitary transformation group. We define a family of Hilbert spaces which are closely related to subconvexity that contain $B^2$ as a dense subset and, as a consequence of the continuity on those spaces of the transformations in $U$, we give a conditional proof of the conjectured asymptotic formula for the sixth moment for every fixed $1/2<\sigma<1$ which, in turn, implies new bounds for the sixth moment on the critical line. We also show that the Lindel\"of hypothesis is a consequence of the continuity of certain more general multiplication operators on those spaces. We conclude with the corollary that the Lindel\"of hypothesis implies that a recent conjecture of Gonek, Hughes and Keating holds in the right-half of the critical strip.
In Chapter \ref{ch4} we consider the general additive divisor problem. Here the divisor functions $d_k(n)$ are the number of ways of writing a natural number $n$ as a product of $k$ factors, and the problem is that of establishing asymptotic formulae for the correlations $\sum_{n\leq x}d_k(n)d_{\ell}(n+h)$ with $h,k,\ell\in\mathbb{N}$. We show that the conjectured asymptotic formulae hold when one or both of the divisor functions are replaced by the minorants $d_k(n,A)=\sum_{m|n,m\leq n^A}d_{k-1}(m)$ with $A$ sufficiently small, leading us to obtain new lower bounds for the asymptotics in the original problem. The main arguments rest on a study of the distribution of the functions $d_k(n,A)$ in arithmetic progressions.
Doctoral Theses
Doctoral College
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