Zeros of Dedekind zeta functions under GRH

TL;DR

This paper establishes explicit bounds on the number and multiplicity of zeros of Dedekind zeta functions within certain imaginary parts under the Generalized Riemann Hypothesis, advancing understanding of their zero distribution.

Contribution

It provides the first explicit bounds on zeros and their multiplicities of Dedekind zeta functions assuming GRH, refining previous qualitative results.

Findings

Explicit upper bounds for zeros in given intervals

Bounds on zero multiplicities under GRH

Enhanced understanding of zero distribution of Dedekind zeta functions

Abstract

Assuming GRH, we prove an explicit upper bound for the number of zeros of a Dedekind zeta function having imaginary part in $[T-a,T+a]$. We also prove a bound for the multiplicity of the zeros.