Average prime-pair counting formula

Jaap Korevaar; Herman te Riele

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

paper

Average prime-pair counting formula

arXiv:0902.4352 · doi:10.1090/S0025-5718-09-02312-6

Abstract

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

26 pages, 6 figures

Average prime-pair counting formula

Abstract

References in corpus (2)

Cited by in corpus (2)