## 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 L_{2}(*x*) - pi_{2}(*x*)
in various intervals up to
*x* = 8 × 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 (integer part of 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 × 10^{10} 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.

*s*_{2} (corresponding to Riemann's approximation
*R(x)*) appears to have mean 0 and standard
deviation *C*.

*s*_{1} (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,

*C*^{2} = 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*((-C*^{2}/2)s).

An equivalent expression for
*C*^{2}/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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