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A method for proving the completeness of a list of zeros of certain L-functions

Author: Jan Büthe
Journal: Math. Comp. 84 (2015), 2413-2431
MSC (2010): Primary 11M26; Secondary 11Y35
Published electronically: February 4, 2015
MathSciNet review: 3356032
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Abstract: When it comes to partial numerical verification of the Riemann Hypothesis, one crucial part is to verify the completeness of a list of pre-computed zeros. Turing developed such a method, based on an explicit version of a theorem of Littlewood on the average of the argument of the Riemann zeta function. In a previous paper by J. Büthe, J. Franke, A. Jost, and T. Kleinjung, we suggested an alternative method based on the Weil-Barner explicit formula. This method asymptotically sacrifices fewer zeros in order to prove the completeness of a list of zeros with imaginary part in a given interval. In this paper, we prove a general version of this method for an extension of the Selberg class including Hecke and Artin L-series, L-functions of modular forms, and, at least in the unramified case, automorphic L-functions. As an example, we further specify this method for Hecke L-series and L-functions of elliptic curves over the rational numbers.

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Jan Büthe
Affiliation: Mathematisches Institut, Bonn University, Endenicher Allee 60, 53115 Bonn, Germany

Received by editor(s): October 28, 2013
Received by editor(s) in revised form: December 2, 2013
Published electronically: February 4, 2015
Article copyright: © Copyright 2015 American Mathematical Society