Missouri Journal of Mathematical Sciences

A Catalog of Interesting Dirichlet Series

H. W. Gould and Temba Shonhiwa

Full-text: Open access

Abstract

A Dirichlet series is a series of the form $$F(s)=\sum_{n=1}^\infty {f(n)\over n^s},$$ where the variable $s$ may be complex or real and $f(n)$ is a number-theoretic function. The sum of the series, $F(s)$, is called the generating function of $f(n)$. The Riemann zeta-function $$\zeta(s)=\sum_{n=1}^{\infty}{1\over n^{s}}=\Pi_{p}\left(1-{1\over p^{s}}\right)^{-1},$$ where $n$ runs through all integers and $p$ runs through all primes is the special case where $f(n) =1$ identically. It is fundamental to the study of prime numbers and many generating functions are combinations of this function. In this paper, we give an overview of some of the commonly known number-theoretic functions together with their corresponding Dirichlet series.

Article information

Source
Missouri J. Math. Sci., Volume 20, Issue 1 (2008), 2-18.

Dates
First available in Project Euclid: 14 September 2011

Permanent link to this document
https://projecteuclid.org/euclid.mjms/1316032830

Digital Object Identifier
doi:10.35834/mjms/1316032830

Mathematical Reviews number (MathSciNet)
MR3589830

Zentralblatt MATH identifier
1143.11005

Subjects
Primary: 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}
Secondary: 11A25: Arithmetic functions; related numbers; inversion formulas

Citation

Gould, H. W.; Shonhiwa, Temba. A Catalog of Interesting Dirichlet Series. Missouri J. Math. Sci. 20 (2008), no. 1, 2--18. doi:10.35834/mjms/1316032830. https://projecteuclid.org/euclid.mjms/1316032830


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