Estimating the number of zeros of Dedekind zeta-functions

TL;DR

This paper introduces a novel method to estimate the count of non-trivial zeros of Dedekind zeta functions up to height T, significantly improving the accuracy of previous asymptotic formulas, including for the Riemann zeta function.

Contribution

It presents a new approach that refines the error term in the zero-counting formula for Dedekind zeta functions, surpassing prior results.

Findings

Improved error bounds for zero-counting asymptotics

Enhanced estimates for the Riemann zeta function zeros

Generalized method applicable to Dedekind zeta functions

Abstract

In this article, I derive a new approach to estimate the number of non-trivial zeros of a given Dedekind zeta function with absolute height at most $T>=1$ counted with multiplicity. The error term in corresponding asymptotic formula improves all previous results, even in the case of the Riemann zeta function.