The Tail of the Singular Series for the Prime Pair and Goldbach Problems

TL;DR

This paper derives an asymptotic formula for a weighted sum related to the tail of the singular series in prime pair and Goldbach problems, advancing understanding of their distribution.

Contribution

It provides a new asymptotic analysis of the tail of the singular series, offering insights into prime pair and Goldbach conjectures.

Findings

Asymptotic formula for the weighted sum of the tail of the singular series.

Improved understanding of the distribution of primes in Goldbach and prime pair problems.

Potential implications for related conjectures in additive prime number theory.

Abstract

We obtain an asymptotic formula for a weighted sum of the square of the tail of the singular series for the Goldbach and prime-pair problems.