Explicit formulae for averages of Goldbach representations

TL;DR

This paper derives explicit formulas for the average number of Goldbach representations using advanced complex analysis techniques, providing new tools for understanding prime sum distributions.

Contribution

It introduces a novel explicit formula for Cesàro-Riesz means of Goldbach representations, extending classical methods with a double Mellin transform approach.

Findings

Explicit formulas for averages of Goldbach representations derived.

Analytic continuation techniques applied to complex functions.

Provides a new framework for studying prime sum representations.

Abstract

We prove an explicit formula, analogous to the classical explicit formula for $\psi(x)$, for the Ces\`aro-Riesz mean of any order $k>0$ of the number of representations of $n$ as a sum of two primes. Our approach is based on a double Mellin transform and the analytic continuation of certain functions arising therein.