Mollification of the Fourth Moment of Dirichlet L-functions

TL;DR

This paper derives an asymptotic formula for the mollified fourth moment of Dirichlet L-functions at the central point for prime moduli, using spectral theory and bounds for Kloosterman sums.

Contribution

It introduces a novel approach to evaluate the mollified fourth moment of Dirichlet L-functions employing spectral analysis and convolution techniques.

Findings

Established an asymptotic formula for the mollified fourth moment

Provided bounds for bilinear forms in Kloosterman sums

Enhanced understanding of non-vanishing of Dirichlet L-functions

Abstract

We evaluate some twisted fourth moment of Dirichlet $L$-functions at the central point s=1/2 and for prime moduli q. The principal tool is a careful analysis of a shifted convolution problem involving the divisor function using spectral theory of automorphic forms and bounds for bilinear forms in Kloosterman sums. Having in mind simultaneous non vanishing results, we apply the Theorem to establish an asymptotic formula of a mollified fourth moment for this family of $L$-functions