Quadratic residues and the distribution of primes
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References
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P. Chebyshev, “Sur une transformation de séries numériques,” Oeuvres, v. 2, 1907, p. 707.
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.
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.
- Edmund Landau, Über einen Satz von Tschebyschef, Math. Ann. 61 (1906), no. 4, 527–550 (German). MR 1511360, DOI 10.1007/BF01449495 E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, v. 2, Chelsea, 1953, p. 701-704. 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. H. Bohr & H. Cramér, “Die Neuere Entwicklung der Analytischen Zahlentheorie,” in Harald Bohr, Collected Mathematical Works, v. 3, Copenhagen, 1952, p. 804. Daniel Shanks, “On the distribution of prime numbers in arithmetic progressions,” Abstract, Goucher Meeting, May 2, 1959 of M. A. A. 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. 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. A. J. C. Cunningham, “Number of primes of given linear forms,” Proc. London Math. Soc., v. 10, series 2, 1911, p. 249-253.
- 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
- John Leech, Note on the distribution of prime numbers, J. London Math. Soc. 32 (1957), 56–58. MR 83001, DOI 10.1112/jlms/s1-32.1.56 G. H. Hardy & Marcel Riesz, The General Theory of Dirichlet’s Series, Cambridge, 1952, p. 3. Ramanujan, in a letter to Hardy, stated that these three classes were “equal". See S. Ramanujan, Collected Papers, Cambridge, 1927, p. 351. 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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Additional Information
- © Copyright 1959 American Mathematical Society
- Journal: Math. Comp. 13 (1959), 272-284
- MSC: Primary 10.00
- DOI: https://doi.org/10.1090/S0025-5718-1959-0108470-8
- MathSciNet review: 0108470