A Gamma Distribution Hypothesis for Prime $k$-tuples

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.

arXiv admin note: substantial text overlap with arXiv:1307.0754

References in corpus (2)

• Evidence of a Gamma Distribution for Prime Powers
• Exact Summatory Functions for Prime $k$-tuples

Cited by in corpus (1)

• A Note on Counting Lattice Points in Bounded Domains