## 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.

### MSC:

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