Average Prime-Pair Counting Formula

摘要

Taking r > 0, let Pi(sub 2r)(x) denote the number of prime pairs (p, p+2r) with p greater than or equal to x. The prime-pair conjecture of Hardy and Littlewood (1923) asserts that Pi(sub 2r)(x) is approximately 2C(sub 2r)li(sub 2)(x) with an explicit constant C(sub 2r) is greater than 0. There seems to be no good conjecture for the remainders w(sub 2r)(x) equals Pi(sub 2r)(x) minus 2C(sub 2r)li(sub 2)(x) that corresponds to Riemann's formula for Pi(x) minus li(x). However, there is a heuristic approximate formula for averages of the remainders w(sub 2r)(x) which is supported by numerical results.