Zeros of partial sums of the Dedekind zeta function of a cyclotomic field

TL;DR

This paper investigates the zeros of truncated Dedekind zeta functions for cyclotomic fields, establishing zero-free regions and estimating the number of zeros up to a certain height.

Contribution

It introduces new zero-free regions and zero count estimates for partial sums of Dedekind zeta functions of cyclotomic fields.

Findings

Zero-free regions for the partial sums are established.

An estimate for the number of zeros up to height T is provided.

Results contribute to understanding the distribution of zeros of truncated Dedekind zeta functions.

Abstract

In this article, we study the zeros of the partial sums of the Dedekind zeta function of a cyclotomic field $K$ defined by the truncated Dirichlet series \[ \zeta_{K, X} (s) = \sum_{\|\mathfrak{a}\| \leq X} \frac{1}{\|\mathfrak{a}\|^{s}}, \] where the sum is to be taken over nonzero integral ideals $\mathfrak{a}$ of $K$ and $\|\mathfrak{a}\|$ denotes the absolute norm of $\mathfrak{a}$. Specifically, we establish the zero-free regions for $\zeta_{K, X} (s)$ and estimate the number of zeros of $\zeta_{K, X} (s)$ up to height $T$.