Hi,
I have a question regarding the universality property of the Riemann zeta-function. I am no expert on this, so I'd be glad for any relevant reference.
First, recall Voronin's remarkable theorem on the Universality of the Riemann zeta-function :
Let $K$ be a compact subset with connected complement lying in the strip ${1/2 < \operatorname{Re}(z)<1}$, and let $f : K \rightarrow \mathbb{C}$ be continuous, holomorphic on the interior of $K$, and zero-free on $K$. Then for each $\epsilon>0$, there exists $t>0$ such that $$\max_{z \in K} |\zeta(z+it)-f(z)|<\epsilon.$$ Even more : the lower density of the set of such $t$'s is positive..!
Note that of course, the hypothesis that the complement of $K$ is connected is essential in the above theorem.
My question is the following :
Is there some sort of (modified) zeta-function universality-like result for compact sets $K$ with disconnected complements? For example, if $\mathbb{C}_\infty \setminus K$ has a finite number of components?
EDIT
Of course I know that a sequence of the form $f_n(z):=\zeta(z+it_n)$ won't work in the case when the complement of $K$ is disconnected (such a sequence cannot approximate uniformly say $1/z$ on an annulus centered at $0$). I'm asking wether there is some sequence of functions, involving the Riemann zeta-function, that could work in this case, and generalize Voronin's Theorem. Note that such functions will necessarily have poles in each component of the complement of $K$.
2nd EDIT
Let me explain what I was looking for here. Basically, I'd like to know if there exists a result of the following form :
Let $K$ be a compact subset whose complement has finitely many components lying in the strip ${1/2 < \operatorname{Re}(z)<1}$, and let $f : K \rightarrow \mathbb{C}$ be continuous, holomorphic on the interior of $K$, and zero-free on $K$. Then for each $\epsilon>0$, there exists...
Here insert some uniform approximation of $f$ on $K$ by a function involving the Riemann zeta-function
Furthermore, in the case when $K$ has connected complement, I would like the above result to reduce to Voronin's Theorem.
In summary, I want to know if there exists a generalization of Voronin's Theorem to compact sets whose complement have finitely many components.
Thank you, Malik
