## Dissertations, Theses, and Capstone Projects

## Date of Degree

9-2024

## Document Type

Dissertation

## Degree Name

Ph.D.

## Program

Mathematics

## Advisor

Louis-Pierre Arguin

## Committee Members

Alexander Gamburd

Matthew Junge

## Subject Categories

Number Theory | Other Mathematics | Probability

## Keywords

probability, number theory, riemann zeta function, central limit theorem

## Abstract

We use the method of Radziwill and Soundararajan to prove Selberg’s central limit theorem for the real part of the logarithm of the Riemann zeta function on the critical line in the multivariate case. This gives an alternate proof of a result of Bourgade. An upshot of the method is to determine a rate of convergence in the sense of the Dudley distance. This is the same rate Selberg claims using the Kolmogorov distance. We also achieve the same rate of convergence in the case of Dirichlet L-functions. Assuming the Riemann hypothesis, we improve the rate of convergence by using an approximation for the logarithm of zeta given by Selberg.

## Recommended Citation

Roberts, Asher, "Limit Theorems for L-functions in Analytic Number Theory" (2024). *CUNY Academic Works.*

https://academicworks.cuny.edu/gc_etds/5913