Moments of L-functions and Liouville-Green method

TL;DR

This paper investigates the distribution of central values of L-functions associated with primitive forms of level one as weight increases, using advanced analytical techniques to establish a lower bound on the proportion exceeding a certain threshold.

Contribution

It introduces a novel combination of the Kuznetsov convolution formula and the Liouville-Green method to analyze L-function values at the central point.

Findings

At least 20% of primitive forms have L-values above a logarithmic threshold.

The methods provide new tools for studying the distribution of L-function values.

The approach can be extended to other families of automorphic forms.

Abstract

We show that the percentage of primitive forms of level one and weight $4k\rightarrow \infty$ for which the associated $L$-function at the central point is no less than $(\log{k})^{-2}$ is at least 20%. The key ingredients of our proof are the Kuznetsov convolution formula and the Liouville-Green method.