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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Quadratic residues and the distribution of primes


Author: Daniel Shanks
Journal: Math. Comp. 13 (1959), 272-284
MSC: Primary 10.00
MathSciNet review: 0108470
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References [Enhancements On Off] (What's this?)

  • [1] P. Chebyshev, ``Sur une transformation de séries numériques,'' Oeuvres, v. 2, 1907, p. 707.
  • [2] P. Chebyshev, ``Lettre de M. le professeur Tchébychev à M. Fuss, sur un nouveau théorème relatif aux nombres premiers dans les formes $ 4n + 1$ et $ 4n + 3$,'' Oeuvres, v. 1, 1899, p. 697-698.
  • [3] E. Phragmén, ``Sur le logarithme intégral et la fonction $ f(x)$ de Riemann,'' Öfversigt af Kongl. Vetenskaps, Akademiens Förhandligar, Stockholm, v. 48, 1891-1892, p. 559-616.
  • [4] Edmund Landau, Über einen Satz von Tschebyschef, Math. Ann. 61 (1906), no. 4, 527–550 (German). MR 1511360, http://dx.doi.org/10.1007/BF01449495
  • [5] E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, v. 2, Chelsea, 1953, p. 701-704.
  • [6] G. H. Hardy & J. E. Littlewood, ``Contributions to the theory of the Riemann zeta function and the theory of the distribution of primes,'' Acta Math., v. 14, 1918, p. 127.
  • [7] H. Bohr & H. Cramér, ``Die Neuere Entwicklung der Analytischen Zahlentheorie,'' in Harald Bohr, Collected Mathematical Works, v. 3, Copenhagen, 1952, p. 804.
  • [8] Daniel Shanks, ``On the distribution of prime numbers in arithmetic progressions,'' Abstract, Goucher Meeting, May 2, 1959 of M. A. A.
  • [9] H. F. Scherk, ``Bemerkungen über die Bildung der Primzahlen aus einander,'' Crelle's Journal, v. 10, 1833, p. 201-208. This table is more inaccurate than accurate.
  • [10] J. W. L. Glaisher, ``Separate enumeration of primes of the form $ 4n + 1$ and the form $ 4n + 3$,'' Proc. Roy. Soc., v. 29, 1879, p. 192-197.
  • [11] A. J. C. Cunningham, ``Number of primes of given linear forms,'' Proc. London Math. Soc., v. 10, series 2, 1911, p. 249-253.
  • [12] Heinrich Tietze, Einige Tabellen zur Verteilung der Primzahlen auf Untergruppen der teilerfremden Restklassen nach gegebenem Modul, Abh. Bayer. Akad. Wiss. Math.-Nat. Abt. (N.F.) 1944 (1944), no. 55, 31 (German). MR 0017310 (8,136g)
  • [13] John Leech, Note on the distribution of prime numbers, J. London Math. Soc. 32 (1957), 56–58. MR 0083001 (18,642d)
  • [14] G. H. Hardy & Marcel Riesz, The General Theory of Dirichlet's Series, Cambridge, 1952, p. 3.
  • [15] Ramanujan, in a letter to Hardy, stated that these three classes were ``equal". See S. Ramanujan, Collected Papers, Cambridge, 1927, p. 351.
  • [16] Columns 1, 2, and 4 of Table 7 agree with H. Tietze, Gelöste und Ungelöste Mathematische Probleme aus Aller und Neuer Zeit, v. 1, Munich, 1949, p. 25-26.
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  • [18] S. Skewes, ``On the difference $ \pi (x) - li(x),(I)$,'' Journal London Math. Soc., v. 8, 1933, p. 278.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1959-0108470-8
PII: S 0025-5718(1959)0108470-8
Article copyright: © Copyright 1959 American Mathematical Society