Dirichlet Series with Periodic Coefficients

In this chapter, we consider Dirichlet series associated with periodic arithmetical functions f, sometimes also called periodic zeta-functions. This class of Dirichlet series includes Dirichlet L-functions, but in general these functions do not have an Euler product; anyway, we shall denote them by L(s, f). Such Dirichlet series are rather simple objects which have the advantage that many computations can be done explicitly. We prove universality for Dirichlet series attached to non-multiplicative periodic functions subject to some side restrictions. This leads to an interesting zero-distribution which is rather different to the one of Dirichlet L-functions. The results of this chapter are due to Steuding [340, 342, 343].