A Generalized Elliott-Halberstam Conjecture Implying the Twin Prime Hypothesis

TL;DR

This paper introduces a generalized conjecture about prime distributions that, if true, would imply the twin prime conjecture, offering new insights into prime gaps and patterns.

Contribution

It proposes the Generalized Elliott-Halberstam Conjecture for Shifted Convolutions (GEH-2), linking prime pair distribution to the twin prime hypothesis.

Findings

GEH-2 implies the twin prime conjecture

Provides heuristic and analytic motivation for GEH-2

Discusses implications for prime gaps and k-tuple patterns

Abstract

We propose a generalization of the Elliott-Halberstam conjecture concerning the distribution of prime pairs in arithmetic progressions. This conjecture, which we call the Generalized Elliott-Halberstam Conjecture for Shifted Convolutions (GEH-2), provides a level of distribution for correlations of the von Mangoldt function. We show that GEH-2 implies the twin prime conjecture, describe heuristic and analytic motivation, and discuss implications for prime gaps and $k$-tuple patterns.