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In this chapter, we consider Dirichlet series associated with periodic arithmetical functions , sometimes also called periodic zeta-functions. This class of Dirichlet series includes Dirichlet -functions, but in general these functions do not have an Euler product; anyway, we shall denote them by (). 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 -functions. The results of this chapter are due to Steuding [340, 342, 343].

In this chapter, we shall prove a conditional joint universality theorem for functions in . universality means that we are concerned with simultaneous uniform approximation, a topic invented by Voronin [362, 364]. Of course, such a result cannot hold for an arbitrary family of -functions: e.g., () and () cannot be jointly universal. The -functions need to be sufficiently independent to possess this joint universality property. We formulate sufficient conditions for a family of -functions in order to be jointly universal and give examples when these conditions are fulfilled; for instance, Dirichlet -functions to pairwise non-equivalent characters (this is an old result of Voronin) or twists of -functions in the Selberg class subject to some condition on uniform distribution.

In this chapter, we shall obtain universality for many classical -functions, including Dedekind zeta-functions as well as Hecke and Artin -functions. Further, we shall briefly discuss the arithmetic axioms in the definition of with respect to the Langlands program. We give only a sketch of the analytic theory of all these -functions and refer to Bump et al. [47] for further details. For details from algebraic number theory we refer to Heilbronn's survey [129], the monographs of Murty and Murty [270], of Neukirch [279], and, last but not least, Stark's article [337].