Average prime-pair counting formula

Source abstract

Taking r > 0 r>0 , let π 2 r ( x ) \pi _{2r}(x) denote the number of prime pairs ( p , p + 2 r ) (p,\,p+2r) with p ≤ x p\le x . The prime-pair conjecture of Hardy and Littlewood (1923) asserts that π 2 r ( x ) ∼ 2 C 2 r l i 2 ( x ) \pi _{2r}(x)\sim 2C_{2r}\,\mathrm {li}_2(x) with an explicit constant C 2 r > 0 C_{2r}>0 . There seems to be no good conjecture for the remainders ω 2 r ( x ) = π 2 r ( x ) − 2 C 2 r l i 2 ( x ) \omega _{2r}(x)=\pi _{2r}(x)- 2C_{2r}\,\mathrm {li}_2(x) that corresponds to Riemann’s formula for π ( x ) − l i ( x ) \pi (x)-\mathrm {li}(x) . However, there is a heuristic approximate formula for averages of the remainders ω 2 r ( x ) \omega _{2r}(x) which is supported by numerical results.