On the fluctuations of Littlewood for primes of the form

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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

**[1]**Richard P. Brent,*Irregularities in the distribution of primes and twin primes*, Math. Comp.**29**(1975), 43–56. Collection of articles dedicated to Derrick Henry Lehmer on the occasion of his seventieth birthday. MR**0369287**, https://doi.org/10.1090/S0025-5718-1975-0369287-1**[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 et ,"*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 and Alfred Brauer,*On the exact number of primes in the arithmetic progressions 4𝑛±1 and 6𝑛±1*, J. Reine Angew. Math.**291**(1977), 23–29. MR**0441892**, https://doi.org/10.1515/crll.1977.291.23**[5]**Richard H. Hudson and Carter Bays,*The mean behavior of primes in arithmetic progressions*, J. Reine Angew. Math.**296**(1977), 80–99. MR**0460261**, https://doi.org/10.1515/crll.1977.296.80**[6]**A. E. Ingham,*The distribution of prime numbers*, Cambridge Tracts in Mathematics and Mathematical Physics, No. 30, Stechert-Hafner, Inc., New York, 1964. MR**0184920****[7]**S. Knapowski and P. Turán,*Comparative prime-number theory. III. Continuation of the study of comparison of the progressions ≡1 𝑚𝑜𝑑𝑘 and ≡𝑙 𝑚𝑜𝑑𝑘*, Acta Math. Acad. Sci. Hungar.**13**(1962), 343–364. MR**0146158**, https://doi.org/10.1007/BF02020798**[8]**S. Knapowski and P. Turán,*Comparative prime-number theory. VII. The problem of sign-changes in the general case*, Acta Math. Acad. Sci. Hungar**14**(1963), 241–250. MR**0156826**, https://doi.org/10.1007/BF01895712**[9]**S. Knapowski and P. Turán,*Further developments in the comparative prime-number theory. I*, Acta Arith.**9**(1964), 23–40. MR**0162771**, https://doi.org/10.4064/aa-9-1-23-40**[10]**S. Knapowski and P. Turán,*Further developments in the comparative prime-number theory. II. A modification of Chebyshev’s assertion*, Acta Arith.**10**(1964), 293–313. MR**0174538**, https://doi.org/10.4064/aa-10-3-293-313**[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.**32**(1957), 56–58. MR**0083001**, https://doi.org/10.1112/jlms/s1-32.1.56**[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. Tables Aids Comput.**13**(1959), 272–284. MR**0108470**, https://doi.org/10.1090/S0025-5718-1959-0108470-8

Retrieve articles in *Mathematics of Computation*
with MSC:
10-04,
10H15

Retrieve articles in all journals with MSC: 10-04, 10H15

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1978-0476615-8

Article copyright:
© Copyright 1978
American Mathematical Society