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On the fluctuations of Littlewood for primes of the form $ 4n\not=1$


Authors: Carter Bays and Richard H. Hudson
Journal: Math. Comp. 32 (1978), 281-286
MSC: Primary 10-04; Secondary 10H15
DOI: https://doi.org/10.1090/S0025-5718-1978-0476615-8
MathSciNet review: 0476615
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Abstract: Let $ {\pi _{b,c}}(x)$ denote the number of primes $ \leqslant x$ which are $ \equiv c\;\pmod b$. Among the first 950,000,000 integers there are only a few thousand integers n with $ {\pi _{4,3}}(n) < {\pi _{4,1}}(n)$. The authors find three new widely spaced regions containing hundreds of millions of such integers; the density of these integers and the spacing of the regions is of some importance because of their intimate connection with the truth or falsity of the analogue of the Riemann hypothesis for $ L(s)$. The discovery that the majority of all Integers n less than $ 2 \times {10^{10}}$ with $ {\pi _{4,3}}(n) < {\pi _{4,1}}(n)$ are the 410,000,000 (consecutive) integers lying between 18,540,000,000 and 18,950,000,000 is a major surprise; results are carefully corroborated and some of the implications are discussed.


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  • [1] RICHARD P. BRENT, Tables Concerning Irregularities in the Distribution of Primes and Twin Primes to $ {10^{11}}$, Computer Centre, Australian National University, Canberra, Australia 1975. MR 0369287 (51:5522)
  • [2] P. L. CHEBYSHEV, "Lettre de M. le professeur Tchébychev à M. Fuss sur un nouveaux théorème rélatif aux nombres premiers contenus dans les formes $ 4n \pm 1$ et $ 4n \pm 3$," Bull. de la Classe Phys. de l'Acad. Imp. des Sciences, St. Petersburg, v. 11, 1853, p. 208.
  • [3] G. H. HARDY & J. E. LITTLEWOOD, "Contributions to the theory of the Riemannzeta function and the theory of the distribution of primes," Acta Math., v. 41, 1917, pp. 119-196.
  • [4] RICHARD H. HUDSON & ALFRED BRAUER, "On the exact number of primes in the arithmetic progressions $ 4n \pm 1$ and $ 6n \pm 1$," J. Reine Angew. Math., v. 281, 1976, pp. 23-29. MR 0441892 (56:283)
  • [5] RICHARD H. HUDSON & CARTER BAYS, "The mean behavior of primes in arithmetic progressions," J. Reine Angew. Math., v. 295, 1977. MR 0460261 (57:255)
  • [6] A. E. INGHAM, The Distribution of Prime Numbers, Stechert-Hafner, New York, 1964. MR 0184920 (32:2391)
  • [7] S. KNAPOWSKI & P. TURÁN, "Comparative prime-number theory. III. Continuation of the study of comparison of the progressions $ \equiv 1\;\pmod k$ and $ \equiv 1\;\pmod k$," Acta Math. Acad. Sci. Hungar., v. 13, 1962, pp. 343-364. MR 0146158 (26:3682c)
  • [8] S. KNAPOWSKI & P. TURÁN, "Comparative prime-number theory. VII. The problem of sign-changes in the general case," Acta Math. Acad. Sci. Hungar., v. 14, 1963, pp. 241-250. MR 0156826 (28:70a)
  • [9] S. KNAPOWSKI & P. TURÁN, "Further developments in the comparative prime-number theory. I, " Acta Arith., v. 9, 1964, pp. 23-40. MR 0162771 (29:75)
  • [10] S. KNAPOWSKI & P. TURÁN, "Further developments in the comparative prime-number theory. II, A modification of Chebyshev's assertion," Acta Arith., v. 10, 1964, pp. 293-313. MR 0174538 (30:4739)
  • [11] E. LANDAU, Handbuch der Lehre von der Verteilung der Primzahlen, Chelsea, New York, 1953.
  • [12] E. LANDAU, "Über einige Ältere vermutungen und Behauptungen in der Primzahltheorie," Math Z., v. 1, 1919, pp. 1-24.
  • [13] JOHN LEECH, "Note on the distribution of prime numbers," J. London Math. Soc., v. 32, 1957, pp. 56-58. MR 0083001 (18:642d)
  • [14] D. H. LEHMER, Personal communication to Richard Hudson from Daniel Shanks (May 24, 1976).
  • [15] J. E. LITTLEWOOD, "Sur la distribution des nombres premiers," Comptes Rendus, v. 158, 1914, pp. 1869-1872.
  • [16] G. PÓLYA, "On polar singularities of power series and of Dirichlet series," Proc. London Math. Soc. (2), v. 33, 1932, pp. 85-101.
  • [17] DANIEL SHANKS, "Quadratic residues and the distribution of primes," Math. Comp., v. 13, 1959, pp. 272-284. MR 0108470 (21:7186)

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DOI: https://doi.org/10.1090/S0025-5718-1978-0476615-8
Article copyright: © Copyright 1978 American Mathematical Society

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