Goldbach Representations with several primes

TL;DR

This paper extends the analysis of Goldbach representations to sums of k primes, providing an asymptotic formula with improved error terms and linking the problem to the Riemann Hypothesis.

Contribution

It generalizes previous results for two primes to any number of primes k and establishes an equivalence between the average order of representations and the Riemann Hypothesis.

Findings

Derived an asymptotic formula for k-prime Goldbach representations

Improved error terms in the asymptotic estimates

Proved the equivalence between average order and the Riemann Hypothesis

Abstract

We study an asymptotic formula for average orders of Goldbach representations of an integer as the sum of k primes. We extend the existing result for k=2 to a general k, for which we obtain a better error term. Moreover, we prove an equivalence between the Riemann Hypothesis and a good average order in this case.