Explicit zero-free regions for Dedekind Zeta functions
Number Theory
2012-01-20 v1
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).
Cite
@article{arxiv.1106.1868,
title = {Explicit zero-free regions for Dedekind Zeta functions},
author = {Habiba Kadiri},
journal= {arXiv preprint arXiv:1106.1868},
year = {2012}
}
Comments
To appear in International Journal of Number Theory