From: Jan Kristian Haugland <jkhaug00@studNOSPAM.hia.no>
Subject: Re: prime pairs
Date: Thu, 04 Jan 2001 00:05:03 +0100
Newsgroups: sci.math
Summary: Counting pairs of twin primes

claudia christern wrote:

> Hi all,
>
> while i know that the prime pair conjecture (the infinitude of the pairs of
> numbers p, p+2 where bothe are prime) is yet to be discovered, i was
> wondering whether anyone has done work on the distribution of these prime
> pairs. Just as primes have a distribution of n/log(n), does anyone have a
> clue as to whether prime pairs also have a characteristic ditribution. Email
> me at benoize@mac.com
>
> Cheers, Benoit Debrock

In my thesis ("Application of Sieve Methods to
Prime Numbers", Oxford 1998), I proved that

   pi_2 (x) < (1+o(1)) k C_2 x / log^2 x

where C_2 = product over all odd primes p of
(1 - 1 / (p-1)^2) = 0.66016... and k = 6.8325.
pi_2 (x) denotes the number of twin prime
pairs (p, p+2) with p <= x and o(1) is a
function that tends to 0 as x -> oo.

It is conjectured that k can be set to 2.
For a proof that k can be set to 8, see
"Sieve Methods" by Halberstam & Richert,
chapters 1 and 3. (That's the place to start
if you want to follow the proofs of later
results.)

--

Jan Kristian Haugland
http://home.hia.no/~jkhaug00