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.

中文翻译：

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算术进展中的主要因素数量

摘要

数值实验表明，在某些算术级数中，有更多的素数因子。格雷格·马丁（Greg Martin）推测函数$ \ sum _ {n \ leq x，n \ equiv 1 \ bmod 4} \ omega（n）-\ sum _ {n \ leq x，n \ equiv 3 \ bmod 4} \ omega（ n）$将获得一个恒定符号，如$ x \ rightarrow \ infty $，其中$ \ omega（n）$是$ n $的不同素数的数量。在本文中，我们证明了$ \ sum _ {n \ leq x} \ chi（n）\ Omega（n）$和$ \ sum _ {n \ leq x} \ chi（n）\ omega（ n）$在某些合理的假设下，其中$ \ chi（n）$是Dirichlet字符，$ \ Omega（n）$是$ n $的素数乘以复数计算。我们的结果为马丁的猜想提供了有力的证据。