Nagoya Mathematical Journal

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

Tomio Kubota and Mariko Yoshida

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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 $\Re s > 1/2$.

Article information

Source
Nagoya Math. J., Volume 163 (2001), 1-11.

Dates
First available in Project Euclid: 27 April 2005

Permanent link to this document
https://projecteuclid.org/euclid.nmj/1114631618

Mathematical Reviews number (MathSciNet)
MR1854386

Zentralblatt MATH identifier
0986.11066

Subjects
Primary: 11N37: Asymptotic results on arithmetic functions
Secondary: 11M41: Other Dirichlet series and zeta functions {For local and global ground fields, see 11R42, 11R52, 11S40, 11S45; for algebro-geometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72}

Citation

Kubota, Tomio; Yoshida, Mariko. A note on the congruent distribution of the number of prime factors of natural numbers. Nagoya Math. J. 163 (2001), 1--11. https://projecteuclid.org/euclid.nmj/1114631618


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References

  • H. M. Edwards, Riemann's zeta function, Academic Press (1974).
  • S. Lang, Algebraic number theory, Addison Wesley (1970).