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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!

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I have heard the following "picture" mentioned in this connection. Suppose the half strip you have mentioned is a wide street heading off to infinity, and you are driving a car along it. Suppose the modulus of the Riemann zeta function is like a ceiling above you. Then, whatever smooth shape you imagine for a portion of this ceiling, you will eventually see it on the way(of course, with prescribed accuracy). –  Anweshi Jan 3 '10 at 4:47
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Here is a fairly recent survey by Matsumoto: math.nagoya-u.ac.jp/~kohjimat/Siauliai06.pdf. It will at least partially answer some of your questions. –  Jonas Meyer Jan 3 '10 at 6:22
    
That survey is really a great reference on the current state of the zeta-function theory! –  efq Jan 5 '10 at 8:08
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The most detailed treatment is here Steudings LNM: Value distribution of L functions. A more elementary treatement is Laurincikas - Limit theorems for the Riemann Zeta function. –  plusepsilon.de Nov 18 '10 at 8:51
    
Marc, thank you for mentioning these two great references! Much appreciated! –  efq Nov 21 '10 at 6:49

3 Answers 3

up vote 5 down vote accepted

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.

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Many thanks! Birkhoff (universal) functions yields indeed much more results, it is good to know the right key word! –  efq Jan 5 '10 at 13:31

Q1) Looks fine to me :-)

Q2) Yes, zeta and Dirichlet L-functions came first. There are examples, I think any function from the Serlberg class satisfies universality. There is a vast conjecture of Linnik to the effect that a lot of Dirichlet series satisfy universality (I don't remember the exact conjecture).

Q3) Yes, here is the general idea. To prove universality of $\zeta(s)$ you prove that (for fixed $\sigma$) the tuple $(\zeta(\sigma + i t), \zeta'(\sigma + i t), ..., \zeta^{(k)}(\sigma + i t))$ is dense in $\mathbb{C}^{k}$. Thus given an analytic $f$, you can approximate the first $k$ terms in the Taylor series expansion of $f(s)$ by the first $k$ terms of the Taylor expansion of $\zeta(s)$ when $s$ is close to $\sigma + i T$ for some $T$ very large. Thus with this approximation being made you expect that $|f(s) - \zeta(s)| $ is uniformly small for $s$ close to $\sigma + i T$ for some $T$ very large. Of course this is not the whole argument, because it didn't use the non-vanishing of $f$, but this is the essential idea. So in general you need the joint density of the derivatives of your function to prove universality for it (but you should look at Matsumoto's survey :-))

Q4) Could be very interesting! It's the first time I hear the question being asked... But remember: I'm not an expert on universality!

I think the people in the field are (at least in part) working on an effective version of universality (i.e given $\varepsilon$ how big a $T = T(\varepsilon)$ you must take? Of course "large enough" but we want to know $T$ explicitly in terms of $\varepsilon$). As a research topic I feel it's something that is possible to work quite fast (i.e without extensive preparation).

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As to (Q3) I believe the original approach of Voronin when proving his universality theorem was rather function-analytical one. The density of the above (k+1)-tuple curve then appears as a corollary. In Matsumoto´s survey, however, one can find the backward approach, deducing the universality theorem from the density result. –  efq Jan 5 '10 at 13:45
    
Anyways, this was not what I meant under "general argument" since it uses the zeta-function as a concrete function obtaining then its universality. I was looking for an argument suggesting the abstract existence of a Birkhoff function given some space of holomorphic non-vanishing functions on some open disk without actually knowing the explicit form of that function. This is what I meant with non-constructive proof of existence. –  efq Jan 5 '10 at 13:47

To (Q4): I am not a specialist in the field of universal functions, but I believe what you are loooking for are rather manifolds of universal functions. Try googling for manifolds of Birkhoff-universal functions. There are definitely plenty of interesting results in that topic.

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