A catalog of interesting Dirichlet series. (English) Zbl 1143.11005

Summary: A Dirichlet series is a series of the form
\[ F(s)= \sum_{n=1}^\infty \frac{f(n)}{n^s}, \]
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
\[ \zeta(s)= \sum_{n=1}^\infty \frac{1}{n^s}= \Pi_p\biggl(1- \frac{1}{p^s}\biggr)^{-1}, \]
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.


11A25 Arithmetic functions; related numbers; inversion formulas
11M41 Other Dirichlet series and zeta functions