Asymptotics of Goldbach Representations

TL;DR

This paper reviews the historical development of the asymptotic analysis of Goldbach representations and their connection to the Riemann Hypothesis, highlighting the role of primes in arithmetic progressions and L-functions.

Contribution

It provides a comprehensive historical overview and discusses the equivalence between Goldbach representations and the Riemann Hypothesis, including prime distributions in arithmetic progressions.

Findings

Goldbach representations are asymptotically linked to the Riemann Hypothesis.

Primes in arithmetic progressions relate to zeros of L-functions.

Special cases show weaker relationships with L-function zeros.

Abstract

We present a historical account of the asymptotics of classical Goldbach representations with special reference to the equivalence with the Riemann Hypothesis. When the primes are chosen from an arithmetic progression comparable but weaker relationships exist with the zeros of L-functions.