Explicit results on the distribution of zeros of Hecke $L$-functions

TL;DR

This paper establishes explicit bounds on the distribution of zeros of Hecke L-functions, which are crucial for deriving bounds on prime ideals in number fields and improving Chebotarev density estimates.

Contribution

It provides the first explicit log-free zero density estimate and explicit zero-repulsion results for Hecke L-functions, advancing analytic number theory tools.

Findings

Proved explicit zero density estimates for Hecke L-functions

Established explicit zero-repulsion phenomena for these functions

Set the stage for improved bounds on prime ideals in number fields

Abstract

We prove an explicit log-free zero density estimate and an explicit version of the zero-repulsion phenomenon of Deuring and Heilbronn for Hecke $L$-functions. In forthcoming work of the second author, these estimates will be used to establish explicit bounds on the least norm of a prime ideal in a congruence class group and improve upon existing explicit bounds for the least norm of a prime ideal in the Chebotarev density theorem.