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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.

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Some calculations related to Riemann’s prime number formula
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by Hans Riesel and Gunnar Göhl PDF
Math. Comp. 24 (1970), 969-983 Request permission

Abstract:

The objective of this paper is to study the relation of the complex zeros of the Riemann zeta function to the distribution of prime numbers. This relation arises from a formula of Riemann, which is studied here by extensive machine calculations. To establish the validity of the computations, reasonable upper bounds for the various errors involved are deduced. The analysis makes use of a formula, (32), which seems to be quite new.
References
    B. Riemann, "Über die Anzahl der Primzahlen unter einer gegebenen Grosse," Monatsh. Königl. Preuss. Akad. Wiss. Berlin, 1859, pp. 671–680; see also: B. Riemann, Gesammelte mathematische Werke und wissenschaftlicher Nachlass, reprint, Dover, New York, 1953, pp. 145–155. MR 14, 610. H. von Mangoldt, "Zu Riemanns Abhandlung ’über die Anzahl der Primzahlen unter einer gegebenen Grosse’," J. Reine Angew. Math., v. 114, 1895, pp. 255–305. E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, 2nd ed., Chelsea, New York, 1953, pp. 333–370. MR 16, 904.
  • A. E. Ingham, The distribution of prime numbers, Cambridge Tracts in Mathematics and Mathematical Physics, No. 30, Stechert-Hafner, Inc., New York, 1964. MR 0184920
    • D. N. Lehmer, List of Prime Numbers from 1 to 10,006,721, Stechert-Hafner, New York, 1956. pp. IX–X.
  • J. -P. Gram, Rapport sur quelques calculs entrepris par M. Bertelsen et concernant les nombres premiers, Acta Math. 17 (1893), no. 1, 301–314 (French). MR 1554842, DOI 10.1007/BF02391997
  • D. H. Lehmer, On the exact number of primes less than a given limit, Illinois J. Math. 3 (1959), 381–388. MR 106883, DOI 10.1215/ijm/1255455259
  • David C. Mapes, Fast method for computing the number of primes less than a given limit, Math. Comp. 17 (1963), 179–185. MR 158508, DOI 10.1090/S0025-5718-1963-0158508-8
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Additional Information
  • © Copyright 1970 American Mathematical Society
  • Journal: Math. Comp. 24 (1970), 969-983
  • MSC: Primary 10.41
  • DOI: https://doi.org/10.1090/S0025-5718-1970-0277489-3
  • MathSciNet review: 0277489