A Gamma Distribution Hypothesis for Prime $k$-tuples

TL;DR

This paper proposes a gamma distribution-based hypothesis for prime $k$-tuples that improves estimates over traditional conjectures and introduces $k$-tuple zeta functions for potential proofs of prime distribution patterns.

Contribution

It introduces a novel gamma distribution hypothesis for prime $k$-tuples and develops associated $k$-tuple zeta functions to advance understanding of prime distributions.

Findings

Provides better estimates for prime $k$-tuple counts.

Defines $k$-tuple zeta functions linked to prime $k$-tuple conjectures.

Suggests a pathway to prove the $k$-tuple prime number theorem.

Abstract

We conjecture average counting functions for prime $k$-tuples based on a gamma distribution hypothesis for prime powers. The conjecture is closely related to the Hardy-Littlewood conjecture for $k$-tuples but yields better estimates. Possessing average counting functions along with their corresponding exact counting functions allows to implicitly define pertinent $k$-tuple zeta functions. The $k$-tuple zeta functions in turn allow construction of $k$-tuple analogs of explicit formulae. If the zeros of the (implicitly defined) $k$-tuple zeta can be determined, the explicit formulae should yield a (dis)proof of the $k$-tuple analog of the prime number theorem.