Is there a general formula that sums up all values of $ζ′(2n)$, such that $n\in\mathbb{N}$?
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$\begingroup$ Other than $-2 \sum_{n=2}^\infty (\log n)n^{-2}/ (1 -n^{-2})^2$? $\endgroup$– Will SawinCommented Mar 21, 2023 at 12:19
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$\begingroup$ Yes. I want close form. $\endgroup$– L.LCommented Mar 21, 2023 at 12:24
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2$\begingroup$ Well, it is a real constant completely determined by its definition. But so are $\pi$ and $e$, so what will change if we find some fancy relation between them? Of course, if it were rational, that would make some difference, but I don't think that this will be proved any time soon and I would rather bet that the number is transcendental. In other words, what exactly are you intending to do with that closed form that cannot be done with the series representation? $\endgroup$– fedjaCommented Mar 21, 2023 at 13:23
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$\begingroup$ I think the general expectation is that there is no such "closed form" ... unless we perhaps allow considerably more exotic notions of "closed form", akin to conjectural things about $\zeta(\mathrm{odd})$ (aiming in a different direction than Apery, et al). $\endgroup$– paul garrettCommented Mar 21, 2023 at 17:12
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$\begingroup$ @fedja Finding closed form expressions for infinite sums is a pretty crucial thing to want to do. You might just have simply asked the OP the motivation for the problem. $\endgroup$– Jesse ElliottCommented Mar 21, 2023 at 22:37
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Partial answer: in my book Springer Graduate Texts in Math 240 page 142 Exercise 92, I give six equivalent formulas for the very similar sum $\sum_{n\ge2}\zeta'(n)$. I suggest first that you prove the equivalence of the six, and second that you adapt this proof to your sum, which should probably also give you six equivalent formulas.
answered Mar 21, 2023 at 21:17
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1$\begingroup$ Totally off topic, but that's a wonderful book that I found very useful to me both in learning number theory and learning to love number theory. $\endgroup$– anomalyCommented Mar 21, 2023 at 22:52
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1$\begingroup$ Just for completeness, the title of the book is "Number Theory: Volume II: Analytic and Modern Tools". $\endgroup$– WojowuCommented Mar 22, 2023 at 1:41