The error term in the Ces\`{a}ro mean of the prime pair singular series

TL;DR

This paper demonstrates that the error term in the Cesàro mean of the prime pair singular series cannot be minimized too much and exhibits oscillatory behavior, impacting the understanding of prime pair distributions.

Contribution

It establishes lower bounds and oscillation properties of the error term in the Cesàro mean of the prime pair singular series, revealing limitations in its approximation.

Findings

Error term cannot be arbitrarily small

Error term exhibits oscillatory behavior

Implications for prime pair conjectures

Abstract

We show that the error term in the asymptotic formula for the Ces{\`a}ro mean of the singular series in the Goldbach and the Hardy-Littlewood prime-pair conjectures cannot be too small and oscillates.