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Are there known relationships linking special values of the Riemann zeta function or MZV (multiple zeta values, i.e. $\zeta(n_1, \cdots n_k)$ with $n_i \in \mathbb Z^+$) to the nontrivial zeroes of the Riemann zeta function?

I became curious of this because MZVs appear in Feynman amplitude calculations. For example, we find a $\zeta(3,5)$ in the $\phi^4$ theory of page 7 of Francis Brown's paper. Also, $\zeta(3), \zeta(5)$ are seen from a quantum electrodynamics calculation for the magnetic dipole moment of a muon, as seen on page 164 of this collaborative effort.

Mysticism should be avoided, but if the zeroes of the Riemann zeta say something about MZVs or just $\zeta(2n+1)$'s, then maybe the distribution of prime numbers says something about Feynman amplitudes.

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  • $\begingroup$ $\zeta(2n+1)$ is expressed explicitly in terms of the zeros via the Hadamard product. What more could you possibly want? $\endgroup$
    – Stopple
    Commented May 3, 2021 at 14:06
  • $\begingroup$ Thanks for the pointer, I simply didn't know about that formula (I'm aware of such a general expansion, but didn't know that it would work for $\zeta$). Is there a similar formula for multiple zeta functions? $\endgroup$
    – Uzu Lim
    Commented May 3, 2021 at 16:40
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    $\begingroup$ According to this paper by Zhao (from 2000) ams.org/journals/proc/2000-128-05/S0002-9939-99-05398-8/… we don't even understand the trivial zeros of multiple zeta functions. I don't know enough about functions of more than one complex variable to know if there is some generalization of Hadamard product. $\endgroup$
    – Stopple
    Commented May 3, 2021 at 20:21

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