Mean representation number of integers as the sum of primes

TL;DR

Under the assumption of the Riemann Hypothesis, the paper derives asymptotic estimates for the average number of ways integers can be expressed as sums of two primes, establishing the optimality of these estimates through an Omega-term.

Contribution

The paper provides the first asymptotic estimates for the mean representation number of integers as sums of two primes under the Riemann Hypothesis, including a proof of the estimate's optimality.

Findings

Asymptotic formulas for mean representation numbers under RH

Proof of the optimality of these asymptotic estimates

Establishment of the best possible bounds through Omega-term

Abstract

Assuming the Riemann Hypothesis we obtain asymptotic estimates for the mean value of the number of representations of an integer as a sum of two primes. By proving a corresponding Omega-term, we prove that our result is essentially the best possible.