IRREGULARITIES IN THE DISTRIBUTION OF PRIMES AND TWIN PRIMES

RICHARD P. BRENT

Dedicated to Derrick H. Lehmer on the occasion of his 70th birthday

Abstract

The maxima and minima of ⟨L(x)⟩−π(x), ⟨R(x)⟩−π(x), and ⟨L_{2}(x)⟩−π_{2}(x) in various intervals up to x = 8×10^{10} are tabulated. Here π(x) and π_{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 π(x), and L_{2}(x) is the Hardy-Littlewood approximation to π_{2}(x). The computation of the sum of inverses of twin primes less than 8 × 10^{10} gives a probable value 1.9021604 ± 5 × 10^{−7} for Brun’s constant.

Comments

Only the Abstract is given here. The full paper appeared as [1]. For a more recent evaluation of Brun’s constant, which incidentally resulted in the discovery of a bug in the Pentium floatingpoint divide, see [3].

Errata. On page 49, three lines from the bottom, “17” should be replaced by “16”, and “900” by “960”. On the same page, six lines from the bottom, “17” should be replaced by “16”.

References

[1] R. P. Brent, “Irregularities in the distribution of primes and twin primes”, Mathematics of Computation (Derrick H. Lehmer special issue) 29 (1975), 43–56. MR 50#1791, 51#5522; Zbl 295.10002. Errata: ibid 30 (1976), 198. MR 53#302. See also “Tables concerning irregularities in the distribution of primes and twin primes”, UMT 4, ibid 29 (1975), 331; and “Tables concerning irregularities in the distribution of primes and twin primes to 10^{11}”, UMT 21, ibid 30 (1976), 379. rpb024.

[2] R. Sherman Lehman, “On the difference π(x) − li(x)”, Acta Arithmetica 11 (1966), 397–410. MR 34#2546.

[3] T. R. Nicely, “Enumeration to 10^{14} of the Twin Primes and Brun’s Constant”, Virginia Journal of Science 46 (1995), 195–204. For a review see review04.dvi.gz .

Computer Centre, Australian National University, Canberra, Australia

1991 Mathematics Subject Classification. Primary 11-04, 11Y60, 11Y70; Secondary 11A41, 65A05, 65B05. Key words and phrases. Prime, twin prime, Riemann’s approximation, error bounds, Hardy-Littlewood con-

jecture, Brun’s constant, logarithmic integral.

The author is indebted to Daniel Shanks for acquainting him with reference [2] and making several very fruitful suggestions.

Copyright \begin{matrix}c \\ ◯ \end{matrix} 1975, American Mathematical Society. Comments \begin{matrix}c \\ ◯ \end{matrix} 1996, R. P. Brent.

rpb024a typeset using AMS-L^{A}TEX.