Average prime-pair counting formula

Korevaar, Jaap; te Riele, Herman

doi:10.1090/S0025-5718-09-02312-6

Average prime-pair counting formula
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by Jaap Korevaar and Herman te Riele PDF

Math. Comp. 79 (2010), 1209-1229 Request permission

Abstract:

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

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