Mollified Moments of Cubic Dirichlet L-Functions over the Eisenstein Field

TL;DR

This paper proves, assuming GRH, that a positive density of cubic Dirichlet L-functions over the Eisenstein field do not vanish at the central point by analyzing mollified moments, with explicit but small non-vanishing proportion.

Contribution

It provides the first mollified moment calculation and sharp bounds for higher moments of these L-functions under GRH, establishing non-vanishing results.

Findings

Positive density of non-vanishing L-functions at s=1/2

Unconditional first mollified moment calculation

Sharp upper bounds for higher mollified moments

Abstract

We prove, assuming the generalized Riemann Hypothesis (GRH) that there is a positive density of $L$-functions associated with primitive cubic Dirichlet characters over the Eisenstein field that do not vanish at the central point $s=1/2$. This is achieved by computing the first mollified moment, which is obtained unconditionally, and finding a sharp upper bound for the higher mollified moments for these $L$-functions, under GRH. The proportion of non-vanishing is explicit, but extremely small.