The error term in counting prime pairs

TL;DR

This paper investigates the error term in the Hardy-Littlewood conjecture for prime pairs, linking it to the L^1 norm of a specific exponential sum over primes involving the von Mangoldt function.

Contribution

It establishes a novel connection between the size of the error term and the L^1 norm of an exponential sum, providing new insights into prime pair distribution.

Findings

Error term size related to exponential sum norm

Connection between Hardy-Littlewood conjecture and exponential sums

Potential implications for prime pair counting accuracy

Abstract

We relate the size of the error term in the Hardy-Littlewood conjectured formula for the number of prime pairs to the $L^{1}$ norm of an exponential sum over the primes formed with the von Mangoldt function.