## Irregularities in the distribution of primes and twin primes

HTML articles powered by AMS MathViewer

- by Richard P. Brent PDF
- Math. Comp.
**29**(1975), 43-56 Request permission

Corrigendum: Math. Comp.

**30**(1976), 198.

## Abstract:

The maxima and minima of $\langle L(x)\rangle - \pi (x),\langle R(x)\rangle - \pi (x)$, and $\langle {L_2}(x)\rangle - {\pi _2}(x)$ in various intervals up to $x = 8 \times {10^{10}}$ are tabulated. Here $\pi (x)$ and ${\pi _2}(x)$ are respectively the number of primes and twin primes not exceeding $x,L(x)$ is the logarithmic integral, $R(x)$ is Riemann’s approximation to $\pi (x)$, and ${L_2}(x)$ is the Hardy-Littlewood approximation to ${\pi _2}(x)$. The computation of the sum of inverses of twin primes less than $8 \times {10^{10}}$ gives a probable value $1.9021604 \pm 5 \times {10^{ - 7}}$ for Brun’s constant.## References

- K. I. APPEL & J. B. ROSSER,
- Patrick Billingsley,
*Prime numbers and Brownian motion*, Amer. Math. Monthly**80**(1973), 1099–1115. MR**345144**, DOI 10.2307/2318544 - Jan Bohman,
*On the number of primes less than a given limit*, Nordisk Tidskr. Informationsbehandling (BIT)**12**(1972), 576–578. MR**321890**, DOI 10.1007/bf01932967 - Jan Bohman,
*Some computational results regarding the prime numbers below $2,000,000,000$*, Nordisk Tidskr. Informationsbehandling (BIT)**13**(1973), 242–244. MR**321852**, DOI 10.1007/bf01933496 - Richard P. Brent,
*The first occurrence of large gaps between successive primes*, Math. Comp.**27**(1973), 959–963. MR**330021**, DOI 10.1090/S0025-5718-1973-0330021-0 - Richard P. Brent,
*The distribution of small gaps between successive primes*, Math. Comp.**28**(1974), 315–324. MR**330017**, DOI 10.1090/S0025-5718-1974-0330017-X
V. BRUN, "La série $1/5 + 1/7 + 1/11 + 1/13 + 1/17 + 1/19 + 1/29 + 1/31 + 1/41 + 1/43 + 1/59 + 1/61 + \ldots$, où les dénominateurs sont ’nombres premiers jumeaux’ est convergente ou finie," - Carl-Erik Fröberg,
*On the prime zeta function*, Nordisk Tidskr. Informationsbehandling (BIT)**8**(1968), 187–202. MR**236123**, DOI 10.1007/bf01933420
G. H. HARDY & J. E. LITTLEWOOD, "Contributions to the theory of the Riemann zeta function and the theory of the distribution of primes," - 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** - R. Sherman Lehman,
*On the difference $\pi (x)-\textrm {li}(x)$*, Acta Arith.**11**(1966), 397–410. MR**202686**, DOI 10.4064/aa-11-4-397-410 - D. H. Lehmer,
*On the exact number of primes less than a given limit*, Illinois J. Math.**3**(1959), 381–388. MR**106883**
J. E. LITTLEWOOD, "Sur la distribution des nombres premiers," - David C. Mapes,
*Fast method for computing the number of primes less than a given limit*, Math. Comp.**17**(1963), 179–185. MR**158508**, DOI 10.1090/S0025-5718-1963-0158508-8 - Ole Møller,
*Quasi double-precision in floating point addition*, Nordisk Tidskr. Informationsbehandling (BIT)**5**(1965), 37–50. MR**181130**, DOI 10.1007/bf01937505 - J. Barkley Rosser and Lowell Schoenfeld,
*Approximate formulas for some functions of prime numbers*, Illinois J. Math.**6**(1962), 64–94. MR**137689** - 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**18688** - Daniel Shanks,
*Quadratic residues and the distribution of primes*, Math. Tables Aids Comput.**13**(1959), 272–284. MR**108470**, DOI 10.1090/S0025-5718-1959-0108470-8
D. SHANKS, UMT - Daniel Shanks and John W. Wrench Jr.,
*Brun’s constant*, Math. Comp.**28**(1974), 293–299; corrigenda, ibid. 28 (1974), 1183. MR**352022**, DOI 10.1090/S0025-5718-1974-0352022-X - S. Skewes,
*On the difference $\pi (x)-\textrm {li}\,x$. II*, Proc. London Math. Soc. (3)**5**(1955), 48–70. MR**67145**, DOI 10.1112/plms/s3-5.1.48 - I. M. Vinogradov,
*A new estimate of the function $\zeta (1+it)$*, Izv. Akad. Nauk SSSR. Ser. Mat.**22**(1958), 161–164 (Russian). MR**0103861**
S. WEINTRAUB, UMT - John W. Wrench Jr.,
*Evaluation of Artin’s constant and the twin-prime constant*, Math. Comp.**15**(1961), 396–398. MR**124305**, DOI 10.1090/S0025-5718-1961-0124305-0

*Table for Functions of Primes*, IDA-CRD Technical Report Number 4, 1961; reviewed in RMT

**55**,

*Math. Comp.*, v. 16, 1962, pp. 500-501.

*Bull. Sci. Math.*, v. 43, 1919, pp. 124-128.

*Acta Math.*, v. 14, 1918, p. 127.

*Comptes Rendus*, v. 158, 1914, pp. 263-266.

**39**,

*Math. Comp.*, v. 17, 1963, p. 307.

**38**,

*Math. Comp.*, v. 27, 1973, pp. 676-677.

## Additional Information

- © Copyright 1975 American Mathematical Society
- Journal: Math. Comp.
**29**(1975), 43-56 - MSC: Primary 10H15; Secondary 10-04
- DOI: https://doi.org/10.1090/S0025-5718-1975-0369287-1
- MathSciNet review: 0369287