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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.78.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


The impact of $\zeta (s)$ complex zeros on $\pi (x)$ for $x<10^{10^{13}}$
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by Douglas A. Stoll and Patrick Demichel PDF
Math. Comp. 80 (2011), 2381-2394 Request permission


An analysis of the local variations of the prime counting function $\pi (x)$ due to the impact of the non-trivial, complex zeros $\varrho _k$ of $\zeta (s)$ is provided for $x<10^{10^{13}}$ using up to 200 billion $\zeta (s)$ complex zeros. A new bound for $|\mathrm {li}(x)-\pi (x)|<x^{1/2}(\mathrm {log} \mathrm {log} \mathrm {log} x+e+1)/e \mathrm {log} x$ is proposed consistent with the error growth rate in Littlewood’s proof that $\mathrm {li}(x)-\pi (x)$ changes sign infinitely often. This bound is also consistent with all presently known cases where $\pi (x)>\mathrm {li}(x)$ including many new examples listed. This implies that Littlewood’s constant $\mathrm {K}=1/e$, the lower bound for Skewes’ number is $3.17\times 10^{114}$ and the positive constant $c$ in the Riemann Hypothesis equivalent $|\mathrm {li}(x)-\pi (x)|<c \mathrm {log}(x)x^{1/2}$ is less than $3\times 10^{-27}$.
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Additional Information
  • Douglas A. Stoll
  • Affiliation: Boeing Research and Technology, Seattle, Washington
  • Email:
  • Patrick Demichel
  • Affiliation: Hewlett-Packard, Les Ulis, France
  • Email:
  • Received by editor(s): November 4, 2009
  • Received by editor(s) in revised form: August 6, 2010
  • Published electronically: April 1, 2011
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Math. Comp. 80 (2011), 2381-2394
  • MSC (2010): Primary 11A41, 11M06, 11M26, 11N05
  • DOI:
  • MathSciNet review: 2813366