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

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]**P. BILLINGSLEY, "Prime numbers and Brownian motion,"*Amer. Math. Monthly*, v. 80, 1973, pp. 1099-1115. MR**0345144 (49:9883)****[3]**J. BOHMAN, "On the number of primes less than a given limit,"*Nordisk Tidskr. Informationsbehandling (BIT)*, v. 12, 1972, pp. 576-577. MR**0321890 (48:255)****[4]**J. BOHMAN, "Some computational results regarding the prime numbers below 2,000,000,000,"*Nordisk Tidskr. Informationsbehandling (BIT)*, v. 13, 1973, pp. 242-244; Errata, ibid., v. 14, 1974, p. 127. MR**0321852 (48:217)****[5]**R. P. BRENT, "The first occurrence of large gaps between successive primes,"*Math. Comp.*, v. 27, 1973, pp. 959-963. MR**0330021 (48:8360)****[6]**R. P. BRENT, "The distribution of small gaps between successive primes,"*Math. Comp.*, v. 28, 1974, pp. 315-324. MR**0330017 (48:8356)****[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]**C.-E. FROBERG, "On the sum of inverses of primes and twin primes,"*Nordisk Tidskr. Informationsbehandling (BIT)*, v. 1, 1961, pp. 15-20. MR**0236123 (38:4421)****[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 Tract, no. 30, Cambridge Univ. Press, New York, 1932. MR**1074573 (91f:11064)****[11]**R. SHERMAN LEHMAN, "On the difference ,"*Acta Arith.*, v. 11, 1966, pp. 397-410. MR**34**#2546. MR**0202686 (34:2546)****[12]**D. H. LEHMER, "On the exact number of primes less than a given limit,"*Illinois J. Math.*, v. 3, 1959, pp. 381-388. MR**21**#5613. MR**0106883 (21:5613)****[13]**J. E. LITTLEWOOD, "Sur la distribution des nombres premiers,"*Comptes Rendus*, v. 158, 1914, pp. 263-266.**[14]**D. C. MAPES, "Fast method for computing the number of primes less than a given limit,"*Math. Comp.*, v. 17, 1963, pp. 179-185. MR**0158508 (28:1731)****[15]**O. MØLLER, "Quasi double-precision in floating point addition,"*Nordisk Tidskr. Informationsbehandling (BIT)*, v. 5, 1965, pp. 37-50; Comment,*ibid.*, v. 5, 1965, pp. 251-255. MR 31 #5359. MR**0181130 (31:5359)****[16]**J. B. ROSSER & L. SCHOENFELD, "Approximate formulas for some functions of prime numbers,"*Illinois J. Math.*, v. 6, 1962, pp. 64-94; reviewed in RMT**40**,*Math. Comp.*, v. 17, 1963, pp. 307-308. MR**25**#1139. MR**0137689 (25:1139)****[17]**E. S. SELMER, "A special summation method in the theory of prime numbers and its application to 'Brun's sum',"*Nordisk Mat. Tidskr.*, v. 24, 1942, pp. 74-81. (Norwegian) MR 8, 316. MR**0018688 (8:316g)****[18]**D. SHANKS, "Quadratic residues and the distribution of primes,"*Math. Tables Aids Comput.*, v. 13, 1959, pp. 272-284. MR**21**#7186. MR**0108470 (21:7186)****[19]**D. SHANKS, UMT**39**,*Math. Comp.*, v. 17, 1963, p. 307.**[20]**D. SHANKS & J. W. WRENCH, JR., "Brun's constant,"*Math Comp.*, v. 28, 1974, pp. 293-299; Corrigendum,*ibid.*, v. 28, 1974, p. 1183. MR**0352022 (50:4510)****[21]**S. SKEWES, "On the difference . II,"*Proc. London Math. Soc. Ser.*(3), v. 5, 1955, pp. 48-70. MR**16**, 676. MR**0067145 (16:676c)****[22]**I. M. VINOGRADOV, "A new estimate of the function ,"*Izv. Akad. Nauk SSSR Ser. Mat.*, v. 22, 1958, pp. 161-164. (Russian) MR**21**#2624. MR**0103861 (21:2624)****[23]**S. WEINTRAUB, UMT**38**,*Math. Comp.*, v. 27, 1973, pp. 676-677.**[24]**J. W. WRENCH, JR., "Evaluation of Artin's constant and the twin-prime constant,"*Math. Comp.*, v. 15, 1961, pp. 396-398. MR**23**#A1619. MR**0124305 (23:A1619)**

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