Statistics of the zeros of $L$-functions and arithmetic correlations

TL;DR

This thesis explores how the statistical properties of zeros of $L$-functions influence arithmetic correlations and fluctuations, particularly regarding prime numbers, by linking these phenomena to number theoretic $L$-functions.

Contribution

It generalizes the understanding of prime number correlations and fluctuations through the lens of $L$-function zero statistics, highlighting universal and non-universal behaviors.

Findings

Implications of $L$-function zero statistics on prime correlations

Generalization of variance in prime number distribution

Connection between universal $L$-function behavior and arithmetic fluctuations

Abstract

This thesis determines some of the implications of non-universal and emergent universal statistics on arithmetic correlations and fluctuations of arithmetic functions, in particular correlations amongst prime numbers and the variance of the expected number of prime numbers over short intervals are generalised by associating these concepts to $L$-functions arising from number theoretic objects.