A Generalized Elliott-Halberstam Conjecture Implying the Twin Prime Hypothesis
Trey Smith

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 -tuple patterns.
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Taxonomy
TopicsAnalytic Number Theory Research · Algebraic Geometry and Number Theory · Advanced Mathematical Identities
