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In this introduction we give some hints for the importance of the Riemann zeta-function for analytic number theory and present first classic results on its amazing value-distribution due to Harald Bohr but also the remarkable universality theorem of Voronin (including a sketch of his proof). Moreover, we introduce Dirichlet -functions and other generalizations of the zeta-function, discuss their relevance in number theory and comment on their value-distribution. For historical details we refer to Narkiewicz's monograph [277] and Schwarz's surveys [317, 318].

In this chapter, we introduce a class of Dirichlet series satisfying several quite natural analytic axioms in addition with two arithmetic conditions, namely, a polynomial Euler product representation and some kind of prime number theorem. The elements of this class will be the main actors in the sequel; however, for some of the later results we do not need to assume all of these axioms. Further, we shall prove mean-value estimates for the Dirichlet series coe.cients of these -functions as well as asymptotic mean-square formulae on vertical lines in the critical strip. These estimates will turn out to be rather useful in later chapters.

In this chapter, we briefly present facts from probability theory which will be used later. These results can be found in the monographs of Billingsley [21, 22], Buldygin [45], Cramèr and Leadbetter [64], Heyer [133], Laurinčikas [186], and Loève [226]. However, there are two exceptions in this crash course in probability theory. In Sect. 3.3 we present Denjoy's heuristic probabilistic argument for the truth of Riemann's hypothesis. Finally, in Sect. 3.7, we introduce the universe for our later studies on universality, the space of analytic functions, and state some of its properties, following Conway [62] and Laurinčikas [186].

In this chapter, we prove a limit theorem dealing with weakly convergent probability measures for -functions from the class in the space of analytic functions. Throughout this chapter, we assume that  ∈ . We remark that we will not make use of axiom (v), so the results hold in a more general context; however, with respect to later applications, there is no need to introduce a further class. We follow the presentation of Laurinčikas [187, 188] (functions in form a subclass of Matsumoto zeta-functions considered herein). Besides, we refer the interested reader to Laurinčikas' survey [185] and his monograph [186].

Now we shall apply the limit theorem from Chap. 4 to derive information on the value-distribution of -functions. Our approach follows Bagchi [9], respectively, the refinements of Laurinčikas [186]. Using the so-called positive density method, introduced by Laurinčikas and Matsumoto [200], we prove a universality theorem for functions ∈ . Here, we shall make use of axiom (v). This result is essentially due to Steuding [345] (under slightly more restrictive conditions).

In 1989, Selberg defined a rather general class of Dirichlet series having an Euler product, analytic continuation and a functional equation of Riemanntype, and formulated some fundamental conjectures concerning them. His aim was to study the value-distribution of linear combinations of -functions. In the meantime, this so-called Selberg class became an important object of research, but still it is not understood very well. In this chapter, we shall investigate universality for functions in the Selberg class. Therefore, we only present results on this class which are related to our studies; for detailed surveys on the Selberg class, we refer to Kaczorowski and Perelli [160], Perelli [290], and M.R. Murty and V.K. Murty [270].

Many beautiful results on the value-distribution of -functions follow from the general theory of Dirichlet series like the Big Picard theorem (see Boas [26] and Mandelbrojt [234]), but more advanced statements can only be proved by exploiting the characterizing properties (the functional equation and the Euler product). In this chapter, we study the distribution of values of Dirichlet series satisfying a Riemann-type functional equation. These results are due to Steuding [346, 347] and their proofs follow in the main part the methods of Levinson [217], Levinson and Montgomery [218], and Nevanlinna theory.

There is an interesting link between universality and the zero-distribution. As we will show in this chapter, the question whether the zeta-function can approximate itself in the right half of the critical strip turns out to be equivalent to the Riemann hypothesis. This reformulation dates back to Bohr [30] who proved its analogue for Dirichlet -functions to non-principal characters. Bagchi [9] extended this result to the Riemann zeta-function. We shall also consider further generalizations.

In this chapter, we shall use ideas from the previous chapter in order to obtain certain effective results on the value-distribution of -functions. The first sections deal with the density of the approximating in universality theorems. The derived upper bounds are due to Steuding [342, 350]. In the following sections explicit estimates for -values in the half-plane of absolute convergence are obtained. These results are due to Girondo and Steuding [100] and rely on a theorem of Rieger [310], resp. a quantified version of Steuding [344], on e.ective inhomogeneous diophantine approximation.

The phenomenon of universality has many interesting consequences. There are the classic results due to Bohr, mentioned in the introduction, as well as extensions of Ostrowski's theorem on the non-existence of algebraic di.erential equations for Dirichlet series (which actually was part of the 18th Hilbert problem). In this chapter, we prove generalizations for universal -functions, we discuss a disproof of a conjecture on certain mean-square estimates due to Ramachandra by Andersson [1], and, finally, we study the value-distribution of linear combinations of shifts of universal Dirichlet series.