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EXPLICIT ZERO-FREE REGIONS FOR DIRICHLET $L$ -FUNCTIONS

Part of: Zeta and $L$-functions: analytic theory

Published online by Cambridge University Press:  03 April 2018

Habiba Kadiri
Habiba Kadiri*
Affiliation:
Department of Mathematics and Computer Science, University of Lethbridge, 4401 University Drive, Lethbridge, Alberta T1K 3M4, Canada email habiba.kadiri@uleth.ca
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Abstract

Let $L(s,\unicode[STIX]{x1D712})$ be the Dirichlet $L$ -function associated to a non-principal primitive character $\unicode[STIX]{x1D712}$ modulo $q$ with $3\leqslant q\leqslant 400\,000$ . We prove a new explicit zero-free region for $L(s,\unicode[STIX]{x1D712})$ : $L(s,\unicode[STIX]{x1D712})$ does not vanish in the region $\mathfrak{Re}\,s\geqslant 1-1/(R\log (q\max (1,|\mathfrak{Im}\,s|)))$ with $R=5.60$ . This improves a result of McCurley where $9.65$ was shown to be an admissible value for $R$ .

MSC classification

Primary: 11M26: Nonreal zeros of $zeta (s)$ and $L(s, chi)$; Riemann and other hypotheses
Secondary: 11M06: $zeta (s)$ and $L(s, chi)$

Information

Type
Research Article
Information
Mathematika , Volume 64 , Issue 2 , 2018 , pp. 445 - 474
Copyright
Copyright © University College London 2018 

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