Correlation of shifted values of $L$-functions in the hyperelliptic ensemble

TL;DR

This paper establishes bounds for the mean values and correlations of quadratic Dirichlet L-functions over hyperelliptic curves in function fields, advancing understanding of their behavior in the large genus limit.

Contribution

It provides new lower and upper bounds for the mean values and derivatives of these L-functions, revealing their correlation structure across different transitions.

Findings

Established lower bounds for mean values of L-functions.

Derived upper bounds for mean values and derivatives.

Revealed correlation patterns of L-functions in the hyperelliptic ensemble.

Abstract

The moments of quadratic Dirichlet $L$-functions over function fields have recently attracted much attention with the work of Andrade and Keating. In this article, we establish lower bounds for the mean values of the product of quadratic Dirichlet $L$-functions associated with hyperelliptic curves of genus $g$ over a fixed finite field $\mathbb{F}_q$ in the large genus limit. By using the idea of A. Florea \cite{FL3}, we also obtain their upper bounds. As a consequence, we find upper bounds of its derivatives. These lower and upper bounds give the correlation of quadratic Dirichlet $L$-functions associated with hyperelliptic curves with different transitions.