Dirichlet Series and Generating Functions

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.