A note on the congruent distribution of the number of prime factors of natural numbers

Abstract

Let $n = p_{1} p_{2} \cdots p_{r}$ be a product of $r$ prime numbers which are not necessarily different. We define then an arithmetic function $\mu_{m}(n)$ by $$\mu_{m}(n) = \rho^{r} \quad (\rho = e^{2\pi i/m}),$$ where $m$ is a natural number. We further define the function $L(s, \mu_{m})$ by the Dirichlet series $$L(s, \mu_{m}) = \sum_{n=1}^{\infty} \frac{\mu_{m}(n)}{n^{s}} = \prod_{p} \Bigl( 1-\frac{\rho}{p^{s}} \Bigr)^{-1} \quad (\Re s > 1), $$ and will show that $L(s, \mu_{m})$, $(m \geq 3)$, has an infinitely many valued analytic continuation into the half plane $\Re s > 1/2$.