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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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An amortized-complexity method to compute the Riemann zeta function
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by Ghaith A. Hiary PDF
Math. Comp. 80 (2011), 1785-1796 Request permission

Abstract:

A practical method to compute the Riemann zeta function is presented. The method can compute $\zeta (1/2+it)$ at any $\lfloor T^{1/4} \rfloor$ points in $[T,T+T^{1/4}]$ using an average time of $T^{1/4+o(1)}$ per point. This is the same complexity as the Odlyzko-Schönhage algorithm over that interval. Although the method far from competes with the Odlyzko-Schönhage algorithm over intervals much longer than $T^{1/4}$, it still has the advantages of being elementary, simple to implement, it does not use the fast Fourier transform or require large amounts of storage space, and its error terms are easy to control. The method has been implemented, and results of timing experiments agree with its theoretical amortized complexity of $T^{1/4+o(1)}$.
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Additional Information
  • Ghaith A. Hiary
  • Affiliation: Institute for advanced Study, 1 Einstein Drive, Princeton, New Jersey 08540
  • Address at time of publication: University of Waterloo, Department of Pure Mathematics, 200 University Avenue W, Waterloo, ON N2L 3G1, Canada
  • MR Author ID: 930454
  • Email: hiaryg@gmail.com
  • Received by editor(s): February 11, 2010
  • Received by editor(s) in revised form: April 28, 2010
  • Published electronically: January 25, 2011
  • Additional Notes: This material is based upon work supported by the National Science Foundation under agreements No. DMS-0757627 (FRG grant) and No. DMS-0635607.
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Math. Comp. 80 (2011), 1785-1796
  • MSC (2000): Primary 11M06, 11Y16; Secondary 68Q25
  • DOI: https://doi.org/10.1090/S0025-5718-2011-02452-X
  • MathSciNet review: 2785479