What is known about rational sums of powers of zeta values?

Asked Oct 01 '20 at 11:25

Active Oct 01 '20 at 11:25

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A fair bit is known about rational zeta series. This includes identities such as $$ \sum_{n=2}^{\infty} [\zeta(n) -1] = 1 $$ and $$\sum_{n=2}^{\infty} \frac{1}{n} [\zeta(n)-1] = 1-\gamma .$$ Many more identities can be found in articles by e.g. Borwein and Adamchik & Srivastava (here).

So far, I have not been able to find identities for series involving powers of zeta values. For instance, I wonder what the series $$\sum_{n=2}^{\infty}[\zeta(n)-1]^{p} $$ amounts to, for some positive integer $p$. Are there any results on these and related types of sums?

asked Oct 01 '20 at 11:25

Max Muller

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@MarcoCantarini This is one of the types of generalizations I'm thinking about – Max Muller Oct 01 '20 at 12:28
I actually asked an analogous question concerning $\sum_{n=2}^{\infty} \left( (\zeta(n))^k -1\right)$. Expanding your summand via the binomial theorem, it would be a linear combination of $(\zeta(n))^p -1, (\zeta(n))^{p-1} -1, ..., -1$. – Varun Vejalla Oct 01 '20 at 12:38
Nice problem. Work on $p=2$ to start with. – GEdgar Oct 01 '20 at 12:41
@GEdgar thank you. I've tried, but to no avail so far. If you have solution, I'd be very interested in reading about it. – Max Muller Oct 01 '20 at 13:49
@VarunVejalla Ah, it seems we have similar interests. Thank you for the link. – Max Muller Oct 01 '20 at 13:51
@GEdgar do you happen to know whether any articles have been written on rational sums of powers of zeta values? – Max Muller Oct 02 '20 at 23:34
A copy of this question is montioned here – zeraoulia rafik Oct 03 '20 at 21:27

What is known about rational sums of powers of zeta values?

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