Counting zeros of Dedekind zeta functions

TL;DR

This paper derives an explicit bound for the number of non-trivial zeros of Dedekind zeta functions of number fields, improving previous estimates by leveraging recent advances in zero counting techniques.

Contribution

It provides a new explicit bound for zeros of Dedekind zeta functions that improves upon prior results using modern zero counting methods.

Findings

Explicit bound for $N_K(T)$ with improved constants

Enhanced accuracy over previous estimates

Application of recent zero counting techniques

Abstract

Given a number field $K$ of degree $n_K$ and with absolute discriminant $d_K$, we obtain an explicit bound for the number $N_K(T)$ of non-trivial zeros (counted with multiplicity), with height at most $T$, of the Dedekind zeta function $\zeta_K(s)$ of $K$. More precisely, we show that for $T \geq 1$, $$ \Big| N_K (T) - \frac{T}{\pi} \log \Big( d_K \Big( \frac{T}{2\pi e}\Big)^{n_K}\Big)\Big| \le 0.228 (\log d_K + n_K \log T) + 23.108 n_K + 4.520, $$ which improves previous results of Kadiri and Ng, and Trudgian. The improvement is based on ideas from the recent work of Bennett $et$ $al.$ on counting zeros of Dirichlet $L$-functions.