Irregularities in the distribution of primes and twin primes

24. R. P. Brent, Irregularities in the distribution of primes and twin primes, Mathematics of Computation 29 (1975), 43-56 (Derrick H. Lehmer special issue). MR 50#1791, 51#5522. Errata: ibid 30 (1976), 198. MR 53#302.

Abstract: dvi (3K), pdf (84K), ps (29K).

Paper: pdf (932K).

Review by Daniel Shanks in Mathematics of Computation 30 (1976), 379: pdf (144K).

Earlier review by Daniel Shanks in Mathematics of Computation 29 (1975), 331: pdf (83K).

Abstract

The maxima and minima of L(x) - pi(x), R(x) - pi(x), and L2(x) - pi2(x) in various intervals up to x = 8 × 1010 are tabulated. Here pi(x) and pi2(x) are respectively the number of primes and twin primes not exceeding x, L(x) is the (integer part of the) logarithmic integral, R(x) is Riemann's approximation to pi(x), and L2(x) is the Hardy-Littlewood approximation to pi2(x). The computation of the sum of inverses of twin primes less than 8 × 1010 gives a probable value of 1.9021604 + 5 × 10-7 for Brun's constant.

Comments

1. For a more recent evaluation of Brun's constant, which incidentally resulted in the discovery of a bug in the Pentium floating-point divide, see my review of a paper by Thomas Nicely.

2. Page 45 mentions an empirical standard deviation (say C) of about 0.21.

s2 (corresponding to Riemann's approximation R(x)) appears to have mean 0 and standard deviation C.
s1 (corresponding to Gauss's approximation L(x)) appears to have mean 1 and standard deviation C.
Thus on average Riemann's approximation R(x) is better than Gauss's approximation L(x).

This contrasts with the fact that, from Littlewood's "Omega" result (page 44), the two approximations are about equally accurate in the worst case.

3. Assuming the Riemann Hypothesis,

C2 = 2 + gamma - ln(4.pi) = 0.0461914...

where gamma is Euler's constant, so

C = 0.214921887978498...

This constant is related to a constant occurring in the Hadamard product formula for the Riemann zeta function zeta(s): the exponential factor in the Hadamard product is exp((-C2/2)s).

An equivalent expression for C2/2 is the sum of the reciprocals of the complex zeros of the Riemann zeta function (summed in the natural order to ensure convergence).

Thanks to Andrey Kulsha, David Broadhurst and Mike Oakes (November 2006) for these observations on the constant C.
For more see http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0611&L=nmbrthry&T=0&P=1333

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".

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