Abstract

Taking r>0, let π2r(x) denote the number of prime pairs (p, p + 2r) withp ≤ x. The prime-pair conjecture of Hardy and Littlewood (1923) asserts that π2r(x) ∼ 2C2r li2(x) with an explicit constant C2r> 0. There seems to be no good conjecture for the remainders ω2r(x) =π2r(x)−2C2r li2(x) that corresponds to Riemann’s formula for π(x)−li(x). However, there is a heuristic approximate formula for averages of the remainders ω2r(x) which is supported by numerical results.

Citations