| dc.description.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 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. | en_GB |
