From: Jan Kristian Haugland 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