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

 

The second-order term in the asymptotic expansion of $ B(x)$


Author: Daniel Shanks
Journal: Math. Comp. 18 (1964), 75-86
MSC: Primary 10.46; Secondary 10.03
MathSciNet review: 0159174
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  • [1] Edmund Landau, ``Uber die Einteilung der positiven ganzen Zahlen in vier Klassen nach der Mindeszahl der zu ihrer additiven Zusammensetzung erforderlichen Quadrate,'' Archiv der Math. und Physik (3), v. 13, 1908, p. 305-312.
  • [2] Edmund Landau, Handbuch der Lehre von der Verteilung der Primzahlen, v. II, Chelsea, N. Y., 1953, p. 643-645, 668-669.
  • [3] G. H. Hardy, Ramanujan, Chelsea, N. Y., 1959, p. 61-62.
  • [4] William Judson LeVeque, Topics in number theory. Vols. 1 and 2, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1956. MR 0080682 (18,283d)
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  • [6] G. K. Stanley, ``Two assertions made by Ramanujan,'' J. London Math. Soc., v. 3, 1928, p. 232-237.
  • [7] G. K. Stanley, ``Corrigenda,'' J. London Math. Soc., v. 4, 1929, p. 32.
  • [8] Daniel Shanks, ``The second-order term in the asymptotic expansion of $ B(x)$,'' Abstract 599-46, Notices, American Math. Soc., v. 10, 1963, p. 261. Errata, p. 377.
  • [9] A. Berger, ``Sur une sommation de quelque séries,'' Nova acta regiae Soc. Sc. Upsaliensis (3), v. 12, 1883, p. 30.
  • [10] M. Lerch, ``Sur quelques formules relative au nombre des classes,'' Bull. des sc. Math. (2), v. 21, 1897, p. 302-303.
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  • [12] Edmund Landau, ``Ueber die zu einem algebraischen zahlkörper gehörige Zetafunction und die Ausdehnung der Tschebyschefschen Primzahlentheorie auf das Problem der Vertheilung der Primideale,'' J. reine Angew. Math., v. 125, 1903, p. 134-136, 176-179.
  • [13] C. F. Gauss, ``De curva lemniscata,'' Werke, v. 3, Göttingen, 1876, p. 414. (The last 6 of the 25 digits here have been corrected in John W. Wrench, Jr., MTE 132, MTAC, v. 3, 1948, p. 202.)
  • [14] Daniel Shanks and John W. Wrench Jr., The calculation of certain Dirichlet series, Math. Comp. 17 (1963), 136–154. MR 0159796 (28 #3012), http://dx.doi.org/10.1090/S0025-5718-1963-0159796-4
  • [15] Daniel Shanks & Larry P. Schmid, ``Variations on a theorem of Landau,'' (to appear).
  • [16] J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64–94. MR 0137689 (25 #1139)
  • [17] Daniel Shanks, Non-linear transformations of divergent and slowly convergent sequences, J. Math. and Phys. 34 (1955), 1–42. MR 0068901 (16,961e)
  • [18] J. Barkley Rosser, Theory and Application of ∫_{0^{𝑧}}𝑒^{-𝑥²}𝑑𝑥 and ∫_{0^{𝑧}}𝑒^{-𝑝²𝑦²}𝑑𝑦∫^{𝑦}₀𝑒^{-𝑥²}𝑑𝑥. Part I. Methods of Computation, Mapleton House, Brooklyn, N. Y., 1948. MR 0027176 (10,267e)

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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1964-0159174-9
PII: S 0025-5718(1964)0159174-9
Article copyright: © Copyright 1964 American Mathematical Society