Irregularities in the distribution of primes and twin primes

Author:
Richard P. Brent

Journal:
Math. Comp. **29** (1975), 43-56

MSC:
Primary 10H15; Secondary 10-04

DOI:
https://doi.org/10.1090/S0025-5718-1975-0369287-1

Corrigendum:
Math. Comp. **30** (1976), 198.

MathSciNet review:
0369287

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The maxima and minima of , and in various intervals up to are tabulated. Here and are respectively the number of primes and twin primes not exceeding is the logarithmic integral, is Riemann's approximation to , and is the Hardy-Littlewood approximation to . The computation of the sum of inverses of twin primes less than gives a probable value for Brun's constant.

**[1]**K. I. APPEL & J. B. ROSSER,*Table for Functions of Primes*, IDA-CRD Technical Report Number 4, 1961; reviewed in RMT**55**,*Math. Comp.*, v. 16, 1962, pp. 500-501.**[2]**Patrick Billingsley,*Prime numbers and Brownian motion*, Amer. Math. Monthly**80**(1973), 1099–1115. MR**0345144**, https://doi.org/10.2307/2318544**[3]**Jan Bohman,*On the number of primes less than a given limit*, Nordisk Tidskr. Informationsbehandling (BIT)**12**(1972), 576–578. MR**0321890****[4]**Jan Bohman,*Some computational results regarding the prime numbers below 2,000,000,000*, Nordisk Tidskr. Informationsbehandling (BIT)**13**(1973), 242–244. MR**0321852****[5]**Richard P. Brent,*The first occurrence of large gaps between successive primes*, Math. Comp.**27**(1973), 959–963. MR**0330021**, https://doi.org/10.1090/S0025-5718-1973-0330021-0**[6]**Richard P. Brent,*The distribution of small gaps between successive primes*, Math. Comp.**28**(1974), 315–324. MR**0330017**, https://doi.org/10.1090/S0025-5718-1974-0330017-X**[7]**V. BRUN, "La série , où les dénominateurs sont 'nombres premiers jumeaux' est convergente ou finie,"*Bull. Sci. Math.*, v. 43, 1919, pp. 124-128.**[8]**Carl-Erik Fröberg,*On the prime zeta function*, Nordisk Tidskr. Informationsbehandling (BIT)**8**(1968), 187–202. MR**0236123****[9]**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.**[10]**A. E. Ingham,*The distribution of prime numbers*, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1990. Reprint of the 1932 original; With a foreword by R. C. Vaughan. MR**1074573****[11]**R. Sherman Lehman,*On the difference 𝜋(𝑥)-𝑙𝑖(𝑥)*, Acta Arith.**11**(1966), 397–410. MR**0202686****[12]**D. H. Lehmer,*On the exact number of primes less than a given limit*, Illinois J. Math.**3**(1959), 381–388. MR**0106883****[13]**J. E. LITTLEWOOD, "Sur la distribution des nombres premiers,"*Comptes Rendus*, v. 158, 1914, pp. 263-266.**[14]**David C. Mapes,*Fast method for computing the number of primes less than a given limit*, Math. Comp.**17**(1963), 179–185. MR**0158508**, https://doi.org/10.1090/S0025-5718-1963-0158508-8**[15]**Ole Møller,*Quasi double-precision in floating point addition*, Nordisk Tidskr. Informations-Behandling**5**(1965), 37–50. MR**0181130****[16]**J. Barkley Rosser and Lowell Schoenfeld,*Approximate formulas for some functions of prime numbers*, Illinois J. Math.**6**(1962), 64–94. MR**0137689****[17]**Ernst S. Selmer,*A special summation method in the theory of prime numbers and its application to “Brun’s sum.”*, Norsk. Mat. Tidsskr.**24**(1942), 74–81 (Norwegian). MR**0018688****[18]**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**[19]**D. SHANKS, UMT**39**,*Math. Comp.*, v. 17, 1963, p. 307.**[20]**Daniel Shanks and John W. Wrench Jr.,*Brun’s constant*, Math. Comp.**28**(1974), 293–299; corrigenda, ibid. 28 (1974), 1183. MR**0352022**, https://doi.org/10.1090/S0025-5718-1974-0352022-X**[21]**S. Skewes,*On the difference 𝜋(𝑥)-𝑙𝑖𝑥. II*, Proc. London Math. Soc. (3)**5**(1955), 48–70. MR**0067145**, https://doi.org/10.1112/plms/s3-5.1.48**[22]**I. M. Vinogradov,*A new estimate of the function 𝜁(1+𝑖𝑡)*, Izv. Akad. Nauk SSSR. Ser. Mat.**22**(1958), 161–164 (Russian). MR**0103861****[23]**S. WEINTRAUB, UMT**38**,*Math. Comp.*, v. 27, 1973, pp. 676-677.**[24]**John W. Wrench Jr.,*Evaluation of Artin’s constant and the twin-prime constant*, Math. Comp.**15**(1961), 396–398. MR**0124305**, https://doi.org/10.1090/S0025-5718-1961-0124305-0

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

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

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1975-0369287-1

Keywords:
Prime,
twin prime,
Riemann's approximation,
error bounds,
Hardy-Littlewood conjecture,
Brun's constant,
logarithmic integral

Article copyright:
© Copyright 1975
American Mathematical Society