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

$\begingroup$ Yes. I want close form. $\endgroup$– L.LCommented Mar 21, 2023 at 12:24

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

$\begingroup$ 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

$\begingroup$ @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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1 Answer
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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.

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

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