NUMBER OF PRIME FACTORS OVER ARITHMETIC PROGRESSIONS. (5th December 2019)

• Record Type: Journal Article
• Authors: Meng, Xianchang
• Is Part Of: Quarterly journal of mathematics. Volume 71:Part 1(2020)
• Page Start: 97
• Page End: 121
• Publication Date: 2019-12-05
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/qmathj/haz040
• Languages: English
• ISSNs: 0033-5606

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.