Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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

Request Permissions   Purchase Content 
 
 

 

Improved results on the Mertens conjecture


Authors: Yannick Saouter and Herman te Riele
Journal: Math. Comp. 83 (2014), 421-433
MSC (2010): Primary 11-04, 11A15, 11M26, 11Y11, 11Y35
DOI: https://doi.org/10.1090/S0025-5718-2013-02716-0
Published electronically: May 28, 2013
MathSciNet review: 3120597
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this article, we study the Mertens conjecture. We revisit and improve the original constructive disproof of János Pintz. We obtain a new lower bound for the minimal counterexample and new numerical results for this conjecture.


References [Enhancements On Off] (What's this?)

  • [Cho95] B.K. Choudhury.
    The Riemann Zeta-function and its derivative.
    Proceedings: Mathematical and Physical Sciences, 450(1940):447-499, Sep. 1995. MR 1356175 (97e:11095)
  • [Dec10] A. Decker.
    Die Widerlegung der Mertens-Vermutung.
    Mathematisches Institut, Mathematisch-Naturewissenschaftliche Fakultät der Rheinischen Friedrich-Willems-Universität Bonn, Oct. 2010.
  • [Kac07] J. Kaczorowski.
    Results on the Möbius function.
    J. London Math. Soc., 75(2):509-521, 2007. MR 2340242 (2008g:11162)
  • [KtR06] T. Kotnik and H.J.J. te Riele.
    The Mertens conjecture revisited.
    In Springer, Lecture Notes in Computer Science, volume 4076, pages 156-167, 2006. MR 2282922 (2007k:11157)
  • [LLL82] A.K. Lenstra, H.W. Lenstra, and L. Lovasz.
    Factoring polynomials with rational coefficients.
    Mathematische Annalen, 261:513-534, 1982. MR 682664 (84a:12002)
  • [Mer97] F. Mertens.
    Über eine zahlentheoretische Funktion.
    Sitzungberichte Akad. Wiss. Wien IIa, (106), 1897.
  • [OtR85] A.M. Odlyzko and H.J.J. te Riele.
    Disproof of the Mertens conjecture.
    J. Reine Angew. Math., (357):138-160, 1985. MR 783538 (86m:11070)
  • [Pin87] J. Pintz.
    An effective disproof of the Mertens conjecture.
    Astérique, (147-148):325-333, 1987. MR 891440 (88f:11091)
  • [RYS69] J.B. Rosser, J.M. Yohe, and L. Schoenfeld.
    Rigorous computation and the zeros of the Riemann zeta-function.
    Information Processing 68, Vol 1: Mathematics, Software, pp. 70-76, North-Holland, Amsterdam, 1969. MR 0258245 (41:2892)
  • [SD10] Y. Saouter and P. Demichel.
    A sharp region where $ \pi (x)-\mathrm {li}(x)$ is positive.
    Math. Comp., 79(272):2395-2405, 2010. MR 2684372 (2011k:11124)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 11-04, 11A15, 11M26, 11Y11, 11Y35

Retrieve articles in all journals with MSC (2010): 11-04, 11A15, 11M26, 11Y11, 11Y35


Additional Information

Yannick Saouter
Affiliation: Institut Telecom Brest, Bretagne
Email: Yannick.Saouter@enst-bretagne.fr

Herman te Riele
Affiliation: CWI, Amsterdam, Netherlands
Email: Herman.te.Riele@cwi.nl

DOI: https://doi.org/10.1090/S0025-5718-2013-02716-0
Received by editor(s): December 14, 2011
Received by editor(s) in revised form: April 26, 2012, and May 9, 2012
Published electronically: May 28, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society