Timeline for Does there exist a rational polynomial $P(x)\in{\mathbb Q}[x]{}$ such that $P(\zeta(s))=\zeta(P(s))$?
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| Mar 26, 2020 at 2:12 | history | edited | GH from MO | CC BY-SA 4.0 |
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| Mar 26, 2020 at 0:31 | comment | added | GH from MO | @SylvainJULIEN: I think there are plenty of continuous and non-continuous functions from $\mathbb{C}$ to $\mathbb{C}$ that commute with $\zeta$. On the other hand, there probably very few holomorphic of meromorphic functions that commute with $\zeta$. At any rate, I stop here a this site is not a discussion board, and also these questions appear to be pretty random (they have little to do with $\zeta$). | |
| Mar 26, 2020 at 0:19 | comment | added | Sylvain JULIEN | But do these functions commute with $\zeta$? | |
| Mar 25, 2020 at 23:52 | comment | added | GH from MO | @SylvainJULIEN: There are also plenty of continuous functions from $\mathbb{C}$ to $\mathbb{C}$ that permute the zeros of $\zeta(s)$, not just the identity and complex conjugation. | |
| Mar 25, 2020 at 23:49 | comment | added | Sylvain JULIEN | And as a permutation is bijective, if $P$ is continuous, it is either the identity or the complex conjugation. | |
| Mar 25, 2020 at 23:48 | comment | added | GH from MO | @SylvainJULIEN: My intuition is that there are plenty of non-continuous functions commuting with $\zeta$, but I don't want to think about it. | |
| Mar 25, 2020 at 23:47 | comment | added | GH from MO | @zeraouliarafik: If $P(\zeta(s))=\zeta(P(s))$ holds at infinitely many distinct points $(s_1,s_2,\dots)$ with a limit point in $\mathbb{C}$, then it also holds for all complex $s\neq 1$ by the uniqueness of analytic continuation. Hence we infer $P(1)=1$ as before. | |
| Mar 25, 2020 at 23:42 | comment | added | Sylvain JULIEN | @zeraoulia rafik: with $s$ in the critical strip, you get that if $z$ is a non trivial zero of $\zeta$, then $P(\zeta(z))=\zeta(P(z))=P(0)$. As this equality doesn't depend on the zero, it means $P$ is a permutation of the multiset of the non trivial zeros of $\zeta$ and thus $P(0)=0$. | |
| Mar 25, 2020 at 23:34 | comment | added | zeraoulia rafik | @GH from MO , would be the same simple proof with s lie in the critical strip ? | |
| Mar 25, 2020 at 23:32 | comment | added | zeraoulia rafik | @SylvainJULIEN, nice idea probably you meant uses of Voronin's universality to show any commuting complex function to R zeta function must be continious | |
| Mar 25, 2020 at 23:29 | comment | added | Sylvain JULIEN | Can your argument be used to show that any complex function commuting to $\zeta$ is necessarily continuous? | |
| Mar 25, 2020 at 23:27 | history | edited | GH from MO | CC BY-SA 4.0 |
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| Mar 25, 2020 at 23:19 | history | answered | GH from MO | CC BY-SA 4.0 |