NUMBER OF PRIME FACTORS OVER ARITHMETIC PROGRESSIONS

Abstract

Abstract Numerical experiments suggest that there are more prime factors in certain arithmetic progressions than others. Greg Martin conjectured that the function $\sum _{n\leq x, n\equiv 1 \bmod 4} \omega (n)-\sum _{n\leq x, n\equiv 3 \bmod 4} \omega (n)$ will attain a constant sign as $x\rightarrow \infty $, where $\omega (n)$ is the number of distinct prime factors of $n$. In this paper, we prove explicit formulas for both $\sum _{n\leq x}\chi (n)\Omega (n)$ and $\sum _{n\leq x}\chi (n)\omega (n)$ under some reasonable assumptions, where $\chi (n)$ is a Dirichlet character and $\Omega (n)$ is the number of prime factors of $n$ counted with multiplicity. Our results give strong evidence for Martin’s conjecture.