# Analytic Number Theory and the Mean Values of L-functions

• George Dickinson

Student thesis: Phd

### Abstract

This thesis focuses on moments of the Riemann zeta-function and Dirichlet $L$-functions. Finding asymptotic formulae for moments of $L$-functions is an important pursuit, as they can be used to reveal information about the distribution of the functions' zeros and bounds on their order of magnitude, as well as having numerous other applications. Moreover, the moments are interesting in themselves, exhibiting a remarkable structure and often fascinating behaviour. The first three parts of this thesis concern Dirichlet $L$-functions, while the final chapter examines a particular moment of the Riemann zeta-function. In Chapter 2, we obtain an asymptotic formula for the twisted second moment of Dirichlet $L$-functions that breaks the half-barrier with an error term that is uniform in both the $q$-aspect and the $t$-aspect. For general coefficients, we demonstrate that it is possible to take the length of the twist up to height $(qT)^{\frac{1}{2} + \frac{1}{66}}$. We then continue on to prove results with longer twists, under certain assumptions. In the next chapter, we demonstrate two applications of the twisted second moment from the first chapter, namely proving a new zero-density result that is strong near the critical line, and a new result on the proportion of critical zeros of Dirichlet $L$-functions. In Chapter 4 we obtain an asymptotic for the twisted fourth moment of Dirichlet $L$-functions, again simultaneously in the $q$ and $t$-aspect. The final chapter investigates the twisted fourth moment of the Riemann zeta-function in vertical arithmetic progressions up the critical line. Li and Radziwi\l{}\l{} have shown that the second moment of $\zeta(s)$ does not behave uniformly over all vertical arithmetic progressions \cite{vertical}. We derive an asymptotic formula for the fourth moment over a certain set of arithmetic progressions that still exhibits this non-uniform behaviour.
Date of Award 31 Dec 2023 English The University of Manchester Marcus Tressl (Supervisor) & Hung Bui (Supervisor)

### Keywords

• Moments
• Dirichlet L-functions
• Riemann zeta-function

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