Lower bounds for negative moments of Dirichlet $L$-functions to a fixed modulus

TL;DR

This paper derives lower bounds for the negative moments of central values of primitive Dirichlet L-functions at a fixed prime modulus, assuming these values do not vanish, extending understanding of their distribution.

Contribution

It provides the first lower bounds for negative moments of Dirichlet L-functions at a fixed prime modulus for all negative real k, under non-vanishing assumptions.

Findings

Established lower bounds for negative moments of Dirichlet L-functions.

Results hold for all real k<0 under non-vanishing assumptions.

Advances the understanding of the distribution of L-values at the central point.

Abstract

We establish lower bounds for the $2k$-th moment of central values of the family of primitive Dirichlet $L$-functions to a fixed prime modulus for all real $k<0$, assuming the non-vanishing of these $L$-values.