Voronin´s Universality Theorem (for the Riemann zeta-Function) according to Wikipedia: Let $U$ be a compact subset of the "critical half-strip" $\{s\in\mathbb{C}:\frac{1}{2}<Re(s)<1\}$ with connected complement. Let $f:U \rightarrow\mathbb{C}$ be continuous and non-vanishing on $U$ and holomorphic on $U^{int}$. Then $\forall\varepsilon >0$ $\exists t=t(\varepsilon)$ $\forall s\in U: |\zeta(s+it)-f(s)|<\varepsilon $. 


(Q1) Is this the accurate statement of Voronin´s Universality Theorem? If so, are there any (recent) **generalisations** of this statement with respect to, say, shape of $U$ or conditions on $f$ ? (If I am not mistaken, the theorem dates back to 1975.)


(Q2) Historically, were the Riemann zeta-function and Dirichlet L-functions the first examples for functions on the complex plane with such "universality"? Are there any examples for functions (on the complex plane) with such properties beyond the theory of zeta- and L-functions?


(Q3) Is there any known general argument why such functions (on $\mathbb{C}$) "must" exist, i.e. in the sense of a non-constructive proof of existence? (with Riemann zeta-function being considered as a proof of existence by construction). 


(Q4) Is anything known about the structure of the class of functions with such universality property, say, on some given strip in the complex plane? 


(Q5) Are there similar examples when dealing with $C^r$-functions from some open subset of $\mathbb{R}^n$ into $\mathbb{R}^m$ ?


Thanks in advance and Happy New Year!



Universality of zeta- and L-functions
Since, I believe, Jonas Meyer provided an answer to Q1, let me just say about the other questions: The concept of universality is much older. It was in fact introduced by Birkhoff, in the case for entire functions, in 1929 (and that is why universal functions are sometimes called Birkhoff functions) "Demonstration d'un theoreme elementaire sur les fonctions entieres." and by Heins, in the case of bounded holomorphic in the unit disk, in 1955.


A possible reference is "Universal functions in several complex variables" by P.S. Chee.


