Counting Prime $k$-tuples

TL;DR

This paper develops exact and average counting functions for prime $k$-tuples, introduces related zeta functions, and uses them to infer properties of $k$-tuple primes, advancing understanding of their distribution.

Contribution

It presents the first exact summatory functions for prime $k$-tuples and introduces $k$-tuple zeta functions, extending classical prime counting methods.

Findings

Exact counting functions for prime $k$-tuples are derived.

Conjectured average summatory functions based on a gamma distribution hypothesis.

Formulation of $k$-tuple explicit formulae using $k$-tuple zeta functions.

Abstract

Exact summatory functions that count the number of prime $k$-tuples up to some cut-off integer are presented. Related summatory $k$-tuple analogs of the first and second Chebyshev functions are then defined. Using a gamma distribution hypothesis for prime powers, associated average summatory functions are conjectured. With exact and average summatory functions in hand, pertinent $k$-tuple zeta functions can be identified, and Perron's formula allows the formulation of $k$-tuple analogs of explicit formulae. The $k$-tuple zeta functions are then used to make some inferences about $k$-tuple primes.