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Dirichlet Series and Generating Functions

  • Paul J. McCarthy
Chapter
  • 395 Downloads
Part of the Universitext book series (UTX)

Abstract

A series of the form
$$ \sum\limits_{{n - 1}}^{\infty } {\frac{{f(n)}}{{{n^s}}}} $$
(*)
where f is an arithmetical function and s is a real variable, is called a Dirichlet series. It will be called the Dirichlet series of f. There exist Dirichlet series such that for all values of s, the series does not converge absolutely (see Exercise 5.1). If the Dirichlet series of f does converge absolutely for some values of s then for those values of s the series determines a function which, as we shall see, serves as a generating function of f.

Keywords

Positive Integer Dirichlet Series Multiplicative Function Arithmetical Function Infinite Product 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag New York Inc. 1986

Authors and Affiliations

  • Paul J. McCarthy
    • 1
  1. 1.Department of MathematicsUniversity of KansasLawrenceUSA

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