Exact Summatory Functions for Prime $k$-tuples

TL;DR

This paper introduces exact summatory functions for counting prime k-tuples up to a certain limit and defines related Chebyshev function analogs, advancing the understanding of prime k-tuple distributions.

Contribution

It provides explicit summatory functions for prime k-tuples and extends classical Chebyshev functions to k-tuple analogs, offering new tools for prime distribution analysis.

Findings

Explicit formulas for prime k-tuple summatory functions

Definition of k-tuple Chebyshev function analogs

Enhanced understanding of prime k-tuple distribution

Abstract

Exact summatory functions that count the number of prime $k$-tuples up to some cut-off integer are presented. Related $k$-tuple analogs of the first and second Chebyshev functions are then defined.