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Of course, we've not found a closed-form solution for the Riemann Zeta function; one that only involves 'elementary functions' (and not recursion, like Riemann's functional equation). However, I'm wondering if we know for certain that this is impossible. If so, how?

In general, is there any way of knowing if it would be possible or not to find such a solution? Or at least, some clues before we waste time checking?

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  • $\begingroup$ Just look up functional equation for the zeta function. $\endgroup$
    – Jacky Chong
    Jan 24, 2020 at 7:38
  • $\begingroup$ This thread may help, $\endgroup$
    – Raymond Manzoni
    Jan 24, 2020 at 7:51
  • $\begingroup$ @RaymondManzoni Thanks. I'm wondering, beyond explicit/manual proof, is there any way to look at an infinite summation and know this for certain, or at least for a set of functions? $\endgroup$
    – Geza Kerecsenyi
    Jan 24, 2020 at 18:15
  • $\begingroup$ @Geza Kerecsenyi: the Risch algorithm is useful if your infinite summation can be rewritten as an indefinite integral. It allows to find the integral using elementary functions (and some additional special functions) or decide that it can't be done this way ("Symbolic Integration Tutorial" by Manuel Bronstein). Specific non-existence proofs by Matthew P Wiener appear here. $\endgroup$
    – Raymond Manzoni
    Jan 24, 2020 at 23:00
  • $\begingroup$ Concerning hypergeometric summation you may read the free book A=B or the book by Koepf. Note that these general results don't prevent some define integrals and series for some specific parameters to admit a "closed form" (think at the usual integral of the gamma function or at $\zeta(2n)$). These parts may be handled in a more artisanal way or with some heavy artillery like the Meijer_G-function by computer algebra: failing doesn't prove anything... $\endgroup$
    – Raymond Manzoni
    Jan 24, 2020 at 23:14

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Zeta function is already considered a closed-form in most contexts. Hurwitz zeta function of certain arguments can be represented as Bernoulli polynomials. We also can represent trigonometric functions, such as tangent via Hurwitz Zeta. In general, Zeta function cannot be reduced to elementary functions on real or complex numbers, but we cannot rule out it can on some other number-like sets.

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  • $\begingroup$ In general, Zeta function cannot be reduced to elementary functions on real or complex numbers is there any reason for this? Thanks for your answer. $\endgroup$
    – Geza Kerecsenyi
    Jan 24, 2020 at 18:11
  • $\begingroup$ @Geza Kerecsenyi the only reason is because we restricted the set of elementary functions in certain way. $\endgroup$
    – Anixx
    Jan 24, 2020 at 18:55

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