On the fluctuations of Littlewood for primes of the form
Authors:
Carter Bays and Richard H. Hudson
Journal:
Math. Comp. 32 (1978), 281286
MSC:
Primary 1004; Secondary 10H15
MathSciNet review:
0476615
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Abstract: Let denote the number of primes which are . Among the first 950,000,000 integers there are only a few thousand integers n with . 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 . The discovery that the majority of all Integers n less than with 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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 [8]
 S. KNAPOWSKI & P. TURÁN, "Comparative primenumber theory. VII. The problem of signchanges in the general case," Acta Math. Acad. Sci. Hungar., v. 14, 1963, pp. 241250. MR 0156826 (28:70a)
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 S. KNAPOWSKI & P. TURÁN, "Further developments in the comparative primenumber theory. I, " Acta Arith., v. 9, 1964, pp. 2340. MR 0162771 (29:75)
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 JOHN LEECH, "Note on the distribution of prime numbers," J. London Math. Soc., v. 32, 1957, pp. 5658. MR 0083001 (18:642d)
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 D. H. LEHMER, Personal communication to Richard Hudson from Daniel Shanks (May 24, 1976).
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 J. E. LITTLEWOOD, "Sur la distribution des nombres premiers," Comptes Rendus, v. 158, 1914, pp. 18691872.
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 G. PÓLYA, "On polar singularities of power series and of Dirichlet series," Proc. London Math. Soc. (2), v. 33, 1932, pp. 85101.
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 DANIEL SHANKS, "Quadratic residues and the distribution of primes," Math. Comp., v. 13, 1959, pp. 272284. MR 0108470 (21:7186)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197804766158
PII:
S 00255718(1978)04766158
Article copyright:
© Copyright 1978
American Mathematical Society
