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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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A method for proving the completeness of a list of zeros of certain L-functions
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by Jan Büthe PDF
Math. Comp. 84 (2015), 2413-2431 Request permission


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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Additional Information
  • Jan Büthe
  • Affiliation: Mathematisches Institut, Bonn University, Endenicher Allee 60, 53115 Bonn, Germany
  • MR Author ID: 1017601
  • Email:
  • Received by editor(s): October 28, 2013
  • Received by editor(s) in revised form: December 2, 2013
  • Published electronically: February 4, 2015
  • © Copyright 2015 American Mathematical Society
  • Journal: Math. Comp. 84 (2015), 2413-2431
  • MSC (2010): Primary 11M26; Secondary 11Y35
  • DOI:
  • MathSciNet review: 3356032