Translator Disclaimer
2001 A note on the congruent distribution of the number of prime factors of natural numbers
Tomio Kubota, Mariko Yoshida
Nagoya Math. J. 163: 1-11 (2001).

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 $\Re s > 1/2$.

Citation

Download Citation

Tomio Kubota. Mariko Yoshida. "A note on the congruent distribution of the number of prime factors of natural numbers." Nagoya Math. J. 163 1 - 11, 2001.

Information

Published: 2001
First available in Project Euclid: 27 April 2005

zbMATH: 0986.11066
MathSciNet: MR1854386

Subjects:
Primary: 11N37
Secondary: 11M41

Rights: Copyright © 2001 Editorial Board, Nagoya Mathematical Journal

JOURNAL ARTICLE
11 PAGES


SHARE
Vol.163 • 2001
Back to Top