Explicit zero-free regions for Dedekind Zeta functions

TL;DR

This paper establishes explicit zero-free regions for Dedekind zeta functions of number fields with large discriminant, improving previous bounds and detailing conditions for the existence and nature of zeros.

Contribution

It provides new explicit bounds for zero-free regions of Dedekind zeta functions, refining earlier results and extending understanding of their zeros for large discriminant fields.

Findings

Zero-free region: Re(s) > 1 - 1/(12.55 log d_K + 9.69 n_K log|Im s| + 3.03 n_K + 58.63)

At most one zero in the region: Re(s) > 1 - 1/(12.74 log d_K), |Im s| < 1

Improved a result of Stark by a factor of 2 on zero bounds

Abstract

Let K be a number field, n_K its degree, and d_K the absolute value of its discriminant. We prove that, if d_K is sufficiently large, then the Dedekind zeta function associated to K has no zeros in the region: Re(s) > 1 - 1/(12.55 log d_K + 9.69 n_K log|Im s| + 3.03 n_K + 58.63) and |Im s| > 1. Moreover, it has at most one zero in the region: Re (s) > 1- 1/(12.74 log d_K) and |Im s| < 1. This zero if it exists is simple and is real. This argument also improves a result of Stark by a factor of 2: there is at most one zero in the region Re (s) > 1 - 1/(2 log d_K) and |Im s| < 1/(2 log d_K).