Explicit zero-free regions for Dedekind Zeta functions

Details Explicit zero-free regions for Dedekind Zeta functions Explicit zero-free regions for Dedekind Zeta functions

2011-06-09

arxiv.org/abs/1106.1868v1

Int. J. of Number Theory 2 Vol. 8, No. 1 (2012) 1-23

Mathematics

Number Theory

To appear in International Journal of Number Theory

Scientific paper

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).

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