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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 $K$.

Thank you, Malik

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

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 result for compact sets $K$ with disconnected complements? For example, if $\mathbb{C}_\infty \setminus K$ has a finite number of components?

I'm asking this because Bagchi's proof of Voronin's Theorem seems to make use of only a special case of Mergelyan's Theorem.

EDIT Since there seems to be a confusion here, let me explain : the "Mergelyan's theorem" I'm referring to is the following :

Let $K$ be a compact set in the plane such that $\mathbb{C}_{\infty} \setminus K$ has finitely many components. Let $S$ be a finite set of points containing one point in each component of the complement of $K$.

Let $f$ be a function continuous on $K$, and holomorphic on the interior of $K$. Then $f$ can be uniformly approximated on $K$ by rational functions with poles in the prescribed set $S$.

  Note that when the complement of $K$ is connected, one can take $S=\{\infty\}$ and we get that $f$ can be uniformly approximated by polynomials. This is the special case that appears in Bagchi's proof of Voronin's Theorem.

Thank you, Malik

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 $K$.

Thank you, Malik

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 zeta-function universality result for compact sets $K$ with disconnected complements? For example, if $\mathbb{C}_\infty \setminus K$ has a finite number of components?

I'm asking this because Bagchi's proof of Voronin's Theorem seems to make use of Mergelyan's Theorem on uniform rational approximation of holomorphic functions, which also holds in the case when $\mathbb{C}_{\infty} \setminus K$ has a finite number of components. Hence perhaps there is some modified version of the universality theorem that would work for compact sets with disconnected complements. Thank you, Malik

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 result for compact sets $K$ with disconnected complements? For example, if $\mathbb{C}_\infty \setminus K$ has a finite number of components?

I'm asking this because Bagchi's proof of Voronin's Theorem seems to make use of only a special case of Mergelyan's Theorem.

EDIT Since there seems to be a confusion here, let me explain : the "Mergelyan's theorem" I'm referring to is the following :

Let $K$ be a compact set in the plane such that $\mathbb{C}_{\infty} \setminus K$ has finitely many components. Let $S$ be a finite set of points containing one point in each component of the complement of $K$.

Let $f$ be a function continuous on $K$, and holomorphic on the interior of $K$. Then $f$ can be uniformly approximated on $K$ by rational functions with poles in the prescribed set $S$.

Note that when the complement of $K$ is connected, one can take $S=\{\infty\}$ and we get that $f$ can be uniformly approximated by polynomials. This is the special case that appears in Bagchi's proof of Voronin's Theorem.

Thank you, Malik