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The famous and remarkable Voronin's universality theorem states:

Theorem (Voronin 75): let $0<r<1/4$ and suppose that $g(s)$ is a nonvanishing continuous function on the disk $ \vert s \vert \leq r$ that is analytic in the interior. Then for any $\epsilon>0$, there exists a positive real number $\tau$ such that: $$max_{\vert s \vert \leq r} \vert \zeta(s+3/4+i \tau)-g(s) \vert <\epsilon. $$

Which practically means that $g(s)$ could be approximated by $\zeta(s)$ for some "high enough" values of $\tau$ - on the right hand side of the critical strip.

The thing is that there is a certain numerical "mystery" with respect to this theorem - when it comes to the question "how high is high enough"? Of course - there are effective analytic studies in various cases. But its worthwhile to pay attention to direct calculation via computer - it turns out to be really "hard" to verify Voronin numerically (in my eyes, the illustration makes the theorem even more impressive).

For instance - let us take the constant function $g(z)=e^{ \pm 3}$.

Question: Is there any estimate on $\tau$ for which Voronin's approximation works for $g(s)=e^{\pm 3}$?

It is important to point out and compare, for instance, the following graph of $\log \vert \zeta(0.75+e^{0.0001 \tau} i) \vert$ for $\tau = 0,...,250000$ in this case:

$ for $\widetilde{\tau} = 0,...,250000$ " />

As you can see - for quite big values - the function still doesn't seem to cross the bound $\pm 3$. So when does Voronin's theorem start to kick in this case?

(It is interesting to note also some implications to zeros of zeta (RH), for instance.)

The famous and remarkable Voronin's universality theorem states:

Theorem (Voronin 75): let $0<r<1/4$ and suppose that $g(s)$ is a nonvanishing continuous function on the disk $ \vert s \vert \leq r$ that is analytic in the interior. Then for any $\epsilon>0$, there exists a positive real number $\tau$ such that: $$max_{\vert s \vert \leq r} \vert \zeta(s+3/4+i \tau)-g(s) \vert <\epsilon. $$

Which practically means that $g(s)$ could be approximated by $\zeta(s)$ for some "high enough" values of $\tau$ - on the right hand side of the critical strip.

The thing is that there is a certain numerical "mystery" with respect to this theorem - when it comes to the question "how high is high enough"? Of course - there are effective analytic studies in various cases. But its worthwhile to pay attention to direct calculation via computer - it turns out to be really "hard" to verify Voronin numerically (in my eyes, the illustration makes the theorem even more impressive).

For instance - let us take the constant function $g(z)=e^{3}$.

Question: Is there any estimate on $\tau$ for which Voronin's approximation works for $g(s)=e^{3}$?

It is important to point out and compare, for instance, the following graph of $\log \vert \zeta(0.75+e^{0.0001 \tau} i) \vert$ for $\tau = 0,...,250000$ in this case:

$ for $\widetilde{\tau} = 0,...,250000$ " />

As you can see - for quite big values - the function still doesn't seem to cross the bound $\pm 3$. So when does Voronin's theorem start to kick in this case?

(It is interesting to note also some implications to zeros of zeta (RH), for instance.)

The famous and remarkable Voronin's universality theorem states:

Theorem (Voronin 75): let $0<r<1/4$ and suppose that $g(s)$ is a nonvanishing continuous function on the disk $ \vert s \vert \leq r$ that is analytic in the interior. Then for any $\epsilon>0$, there exists a positive real number $\tau$ such that: $$max_{\vert s \vert \leq r} \vert \zeta(s+3/4+i \tau)-g(s) \vert <\epsilon. $$

Which practically means that $g(s)$ could be approximated by $\zeta(s)$ for some "high enough" values of $\tau$ - on the right hand side of the critical strip.

The thing is that there is a certain "mystery" with respect to this theorem - when it comes to the question "how high is high enough"? Of course - one can give some analytic study trying to give an effective result. But nowadays we can also try to directly evaluate and verify it with a computer - the thing is that it turns out to be really hard to do that.

For instance - let us take the constant function $g(z)=e^{ \pm 3}$.

Question: Is there any estimate on $\tau$ for which Voronin's approximation works for $g(s)=e^{\pm 3}$?

It is important point out and compare, for instance, the following graph of $\log \vert \zeta(0.75+e^{0.0001 \tau} i) \vert$ for $\tau = 0,...,250000$ in this case:

$ for $\widetilde{\tau} = 0,...,250000$ " />

As you can see - for quite big values - the function still doesn't seem to cross the bound $\pm 3$. So when does Voronin's theorem start to kick in?

(Note also direct implication to zeros of zeta - RH, for instance.)

The famous and remarkable Voronin's universality theorem states:

Theorem (Voronin 75): let $0<r<1/4$ and suppose that $g(s)$ is a nonvanishing continuous function on the disk $ \vert s \vert \leq r$ that is analytic in the interior. Then for any $\epsilon>0$, there exists a positive real number $\tau$ such that: $$max_{\vert s \vert \leq r} \vert \zeta(s+3/4+i \tau)-g(s) \vert <\epsilon. $$

Which practically means that $g(s)$ could be approximated by $\zeta(s)$ for some "high enough" values of $\tau$ - on the right hand side of the critical strip.

The thing is that there is a certain numerical "mystery" with respect to this theorem - when it comes to the question "how high is high enough"? Of course - there are effective analytic studies in various cases. But its worthwhile to pay attention to direct calculation via computer - it turns out to be really "hard" to verify Voronin numerically (in my eyes, the illustration makes the theorem even more impressive).

For instance - let us take the constant function $g(z)=e^{ \pm 3}$.

Question: Is there any estimate on $\tau$ for which Voronin's approximation works for $g(s)=e^{\pm 3}$?

It is important to point out and compare, for instance, the following graph of $\log \vert \zeta(0.75+e^{0.0001 \tau} i) \vert$ for $\tau = 0,...,250000$ in this case:

$ for $\widetilde{\tau} = 0,...,250000$ " />

As you can see - for quite big values - the function still doesn't seem to cross the bound $\pm 3$. So when does Voronin's theorem start to kick in this case?

(It is interesting to note also some implications to zeros of zeta (RH), for instance.)

The famous and remarkable Voronin's universality theorem states:

Theorem (Voronin 75): let $0<r<1/4$ and suppose that $g(s)$ is a nonvanishing continuous function on the disk $ \vert s \vert \leq r$ that is analytic in the interior. Then for any $\epsilon>0$, there exists a positive real number $\tau$ such that: $$max_{\vert s \vert \leq r} \vert \zeta(s+3/4+i \tau)-g(s) \vert <\epsilon. $$

Which practically means that $g(s)$ could be approximated by $\zeta(s)$ for some "high enough" values of $\tau$ - on the right hand side of the critical strip.

The thing is that there is a certain "mystery" with respect to this theorem - when it comes to the question "how high is high enough"? Of course - one can give some analytic study trying to give an effective result. But nowadays we can also try to directly evaluate and verify it with a computer - the thing is that it turns out to be really hard to do that.

For instance - let us take the constant function $g(z)=e^{ \pm 3}$.

Question: Is there any estimate on $\tau$ for which Voronin's approximation works for $g(s)=e^{\pm 3}$?

It is important point out and compare, for instance, the following graph of $log \vert \zeta(0.75+e^{0.0001 \tau} i) \vert$ for $\tau = 0,...,250000$ in this case:

$ for $\widetilde{\tau} = 0,...,250000$ " />

As you can see - for quite big values - the function still doesn't seem to cross the bound $\pm 3$. So when does Voronin's theorem start to kick in?

(Note also direct implication to zeros of zeta - RH, for instance.)

The famous and remarkable Voronin's universality theorem states:

Theorem (Voronin 75): let $0<r<1/4$ and suppose that $g(s)$ is a nonvanishing continuous function on the disk $ \vert s \vert \leq r$ that is analytic in the interior. Then for any $\epsilon>0$, there exists a positive real number $\tau$ such that: $$max_{\vert s \vert \leq r} \vert \zeta(s+3/4+i \tau)-g(s) \vert <\epsilon. $$

Which practically means that $g(s)$ could be approximated by $\zeta(s)$ for some "high enough" values of $\tau$ - on the right hand side of the critical strip.

The thing is that there is a certain "mystery" with respect to this theorem - when it comes to the question "how high is high enough"? Of course - one can give some analytic study trying to give an effective result. But nowadays we can also try to directly evaluate and verify it with a computer - the thing is that it turns out to be really hard to do that.

For instance - let us take the constant function $g(z)=e^{ \pm 3}$.

Question: Is there any estimate on $\tau$ for which Voronin's approximation works for $g(s)=e^{\pm 3}$?

It is important point out and compare, for instance, the following graph of $\log \vert \zeta(0.75+e^{0.0001 \tau} i) \vert$ for $\tau = 0,...,250000$ in this case:

$ for $\widetilde{\tau} = 0,...,250000$ " />

As you can see - for quite big values - the function still doesn't seem to cross the bound $\pm 3$. So when does Voronin's theorem start to kick in?

(Note also direct implication to zeros of zeta - RH, for instance.)