Lower bound for the remainder in the prime-pair conjecture

TL;DR

This paper investigates the lower bounds of the error term in the prime-pair conjecture, linking it to the zeros of the Riemann zeta function and proposing an approximation analogous to Riemann's for prime counting.

Contribution

It establishes a heuristic connection between the remainder in prime-pair counts and the zeros of the zeta function, providing insights into the conjecture's limitations.

Findings

Heuristic argument suggests the remainder cannot be smaller than x^beta.

Proposes an approximation for pi_{2r}(x) similar to Riemann's for pi(x).

Links the size of the error term to the supremum of zeta zeros' real parts.

Abstract

For any positive integer r, let pi_{2r}(x) denote the number of prime pairs (p, p+2r) with p not exceeding (large) x. According to the prime-pair conjecture of Hardy and Littlewood, pi_{2r}(x) should be asymptotic to 2C_{2r}li_2(x) with an explicit positive constant C_{2r}. A heuristic argument indicates that the remainder e_{2r}(x) in this approximation cannot be of lower order than x^beta, where beta is the supremum of the real parts of zeta's zeros. The argument also suggests an approximation for pi_{2r}(x) similar to one of Riemann for pi(x).