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February 2008 A Catalog of Interesting Dirichlet Series
H. W. Gould, Temba Shonhiwa
Missouri J. Math. Sci. 20(1): 2-18 (February 2008). DOI: 10.35834/mjms/1316032830

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.

Citation

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H. W. Gould. Temba Shonhiwa. "A Catalog of Interesting Dirichlet Series." Missouri J. Math. Sci. 20 (1) 2 - 18, February 2008. https://doi.org/10.35834/mjms/1316032830

Information

Published: February 2008
First available in Project Euclid: 14 September 2011

zbMATH: 1143.11005
MathSciNet: MR3589830
Digital Object Identifier: 10.35834/mjms/1316032830

Subjects:
Primary: 11M41
Secondary: 11A25

Rights: Copyright © 2008 Central Missouri State University, Department of Mathematics and Computer Science

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Vol.20 • No. 1 • February 2008
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