Summatory function of the number of prime factors

TL;DR

This paper investigates the behavior of the sum of the number of prime factors in integers up to x across different arithmetic progressions, providing evidence for a conjecture about their differences and deriving a general theorem for related functions.

Contribution

It offers strong evidence supporting Greg Martin's conjecture and introduces a general theorem for arithmetic functions from the Selberg class.

Findings

Numerical experiments support the conjecture.

Differences in summatory functions tend to have a constant sign for large x.

A general theorem for arithmetic functions from the Selberg class is established.

Abstract

We consider the summatory function of the number of prime factors for integers $\leq x$ over arithmetic progressions. Numerical experiments suggest that some arithmetic progressions consist more number of prime factors than others. Greg Martin conjectured that the difference of the summatory functions should attain a constant sign for all sufficiently large $x$. In this paper, we provide strong evidence for Greg Martin's conjecture. Moreover, we derive a general theorem for arithmetic functions from the Selberg class.