Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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



Some calculations related to Riemann's prime number formula

Authors: Hans Riesel and Gunnar Göhl
Journal: Math. Comp. 24 (1970), 969-983
MSC: Primary 10.41
MathSciNet review: 0277489
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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

  • [1] 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.
  • [2] 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.
  • [3] E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, 2nd ed., Chelsea, New York, 1953, pp. 333-370. MR 16, 904.
  • [4] A. E. Ingham, The Distribution of Prime Numbers, Cambridge Tracts in Math. and Math. Phys., no. 30, Stechert-Hafner, New York, 1964. MR 32 #2391. MR 0184920 (32:2391)
  • [5] D. N. Lehmer, List of Prime Numbers from 1 to 10,006,721, Stechert-Hafner, New York, 1956. pp. IX-X.
  • [6] J. P. Gram, "Rapport sur quelques calculs entrepris par M. Bertelsen et concernant les nombres premiers," Acta Math., v. 17, 1893, pp. 301-314. MR 1554842
  • [7] D. H. Lehmer, "On the exact number of primes less than a given limit," Illinois J. Math., v. 3, 1959, pp. 381-388. MR 21 #5613. MR 0106883 (21:5613)
  • [8] D. C. Mapes, "Fast method for computing the number of primes less than a given limit." Math. Comp., v. 17, 1963, pp. 179-185. MR 28 #1731. MR 0158508 (28:1731)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 10.41

Retrieve articles in all journals with MSC: 10.41

Additional Information

Keywords: Riemann's prime number formula, zeros of Riemann zeta function, Gram's series
Article copyright: © Copyright 1970 American Mathematical Society

American Mathematical Society