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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
DOI: https://doi.org/10.1090/S0025-5718-2015-02922-6
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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  • [Bar81] Klaus Barner, On A. Weil's explicit formula, J. Reine Angew. Math. 323 (1981), 139-152. MR 611448 (82i:12014), https://doi.org/10.1515/crll.1981.323.139
  • [BFJK13] J. Büthe, Jens Franke, Alexander Jost, and Thorsten Kleinjung, Some applications of the Weil-Barner explicit formula, Math. Nachr. 286 (2013), no. 5-6, 536-549. MR 3048130, https://doi.org/10.1002/mana.201100218
  • [BJ10] J. Büthe and A. Jost, Algorithmic Applications of Weil's explicit Formula, 2010, Diplomarbeit.
  • [Boo06] Andrew R. Booker, Artin's conjecture, Turing's method, and the Riemann hypothesis, Experiment. Math. 15 (2006), no. 4, 385-407. MR 2293591 (2007k:11084)
  • [Bre79] Richard P. Brent, On the zeros of the Riemann zeta function in the critical strip, Math. Comp. 33 (1979), no. 148, 1361-1372. MR 537983 (80g:10033), https://doi.org/10.2307/2006473
  • [JL94] Jay Jorgenson, Serge Lang, and Dorian Goldfeld, Explicit Formulas, Lecture Notes in Mathematics, vol. 1593, Springer-Verlag, Berlin, 1994. MR 1329730 (96f:11110)
  • [Leh70] R. Sherman Lehman, On the distribution of zeros of the Riemann zeta-function, Proc. London Math. Soc. (3) 20 (1970), 303-320. MR 0258768 (41 #3414)
  • [Mor77] C. J. Moreno, Explicit formulas in the theory of automorphic forms, Number Theory Day (Proc. Conf., Rockefeller Univ., New York, 1976), Springer, Berlin, 1977, pp. 73-216, Lecture Notes in Math., Vol. 626. MR 0476650 (57 #16209)
  • [Rem90] R. Remmert, Funktionentheorie 2, Springer, 1990, first edition.
  • [Rum93] Robert Rumely, Numerical computations concerning the ERH, Math. Comp. 61 (1993), no. 203, 415-440, S17-S23. MR 1195435 (94b:11085), https://doi.org/10.2307/2152965
  • [Sel92] Atle Selberg, Old and new conjectures and results about a class of Dirichlet series, Proceedings of the Amalfi Conference on Analytic Number Theory (Maiori, 1989), Univ. Salerno, Salerno, 1992, pp. 367-385. MR 1220477 (94f:11085)
  • [Tol97] Emmanuel Tollis, Zeros of Dedekind zeta functions in the critical strip, Math. Comp. 66 (1997), no. 219, 1295-1321. MR 1423079 (98d:11140), https://doi.org/10.1090/S0025-5718-97-00871-5
  • [Tru11a] T. Trudgian, Improvements to Turing's method, Math. Comp. 82 (2011), no. 278, pp. 1053-1061. MR 2813359
  • [Tru11b] Timothy Trudgian, On the success and failure of Gram's law and the Rosser rule, Acta Arith. 148 (2011), no. 3, 225-256. MR 2794929 (2012f:11169), https://doi.org/10.4064/aa148-3-2
  • [Tru14] Timothy S. Trudgian, An improved upper bound for the argument of the Riemann zeta-function on the critical line. II, J. Number Theory 134 (2014), 280-292. MR 3111568, https://doi.org/10.1016/j.jnt.2013.07.017
  • [Tur53] A. M. Turing, Some calculations of the Riemann zeta-function, Proc. London Math. Soc. (3) 3 (1953), 99-117. MR 0055785 (14,1126e)
  • [Wei52] André Weil, Sur les ``formules explicites'' de la théorie des nombres premiers, Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] 1952 (1952), no. Tome Supplementaire, 252-265 (French). MR 0053152 (14,727e)
  • [Wei72] André Weil, Sur les formules explicites de la théorie des nombres, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 3-18 (French, with Russian summary). MR 0379440 (52 #345)

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Additional Information

Jan Büthe
Affiliation: Mathematisches Institut, Bonn University, Endenicher Allee 60, 53115 Bonn, Germany
Email: jbuethe@math.uni-bonn.de

DOI: https://doi.org/10.1090/S0025-5718-2015-02922-6
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

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