THE DISTRIBUTION OF SMALL GAPS BETWEEN SUCCESSIVE PRIMES

RICHARD P. BRENT

Abstract

For r ≥ 1 and large N, a well-known conjecture of Hardy and Littlewood implies that the number of primes p ≤ N such that p + 2r is the least prime greater than p is asymptotic to

∫_{2}^{N} (∑_{k=1}^{r} \frac{A_{r,k}}{(log x)^{k+1}} ) dx ,

where the A_{r,k} are certain constants. We describe a method for computing these constants. Related constants are given to 10D for r = 1(1)40, and some empirical evidence supporting the conjecture is mentioned.

Comments

Only the Abstract is given here. A preliminary version appeared as [1] and the full paper appeared as [2]. Related tables [3] were deposited in the Mathematics of Computation UMT file.

References

[1] R. P. Brent, “Empirical evidence for a proposed distribution of small prime gaps”, Report TR CS 123, Department of Computer Science, Stanford University, February 1969, 18 pp.

[2] R. P. Brent, “The distribution of small gaps between successive primes”, Mathematics of Computation 28 (1974), 315–324. MR 48#8356, Zbl 274.10001. rpb021

[3] R. P. Brent, “The distribution of prime gaps in intervals up to 10^{16}”, UMT 7, Mathematics of Computation 28 (1974), 331–332 (reviewed by Daniel Shanks). rpb021r.

Computer Centre, Australian National University, Canberra, Australia

1991 Mathematics Subject Classification. Primary 10-04, 10A20, 10A25, 10A40, 65A05. Key words and phrases. Prime, distribution of primes, Hardy-Littlewood conjecture, prime gap, twin primes. Copyright \begin{matrix}c \\ ◯ \end{matrix} 1974, American Mathematical Society. Comments \begin{matrix}c \\ ◯ \end{matrix} 2000, R. P. Brent.

rpb021a typeset using AMS-L^{A}TEX.