Average prime-pair counting formula

TL;DR

This paper discusses the prime-pair counting function, the Hardy-Littlewood conjecture, and introduces a heuristic formula for the average behavior of the remainders, supported by numerical evidence.

Contribution

It proposes a heuristic approximate formula for the average of prime-pair counting remainders, filling a gap in understanding their distribution.

Findings

Numerical results support the heuristic formula.

The formula provides insight into the average behavior of prime pairs.

No precise conjecture exists for individual remainders, but averages can be approximated.

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 $\pi_{2r}(x)\sim 2C_{2r} {\rm li}_2(x)$ with an explicit constant $C_{2r}>0$. There seems to be no good conjecture for the remainders $\omega_{2r}(x)=\pi_{2r}(x)- 2C_{2r} {\rm li}_2(x)$ that corresponds to Riemann's formula for $\pi(x)-{\rm li}(x)$. However, there is a heuristic approximate formula for averages of the remainders $\omega_{2r}(x)$ which is supported by numerical results.