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I've been trying to find an extensive, in-depth treatment of rational zeta series. Via the Wikipedia article on the topic, I've found two articles on this subject. While they are certainly very informative, I wonder whether there is a more comprehensive treatment that -- among perhaps other things -- provides an introduction on the summation methods, summarizes all the known results to date, shows what the open problems in the domain are, and shows how the summation of zeta terms relates to finding the sum of terms of other (Dirichlet) series.

I've asked a related question on MSE. The question above is a slightly modified and somewhat more elaborate version of it.

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    $\begingroup$ A lot of analytic number theory textbooks does what you are asking for. Perhaps the best one available currently is “Analytic Number Theory” by Iwaniec and Kowalski. $\endgroup$ May 24, 2020 at 13:03
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    $\begingroup$ @StanleyYaoXiao Hi Stanley, thank you for your suggestion. I've looked through the table of contents of the book you mentioned, but I haven't been able to find a (sub)chapter on rational zeta series. Could you please tell me in which section(s) of the book they write about rational zeta series? $\endgroup$
    – Max Muller
    May 24, 2020 at 13:22
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    $\begingroup$ There is a book by Srivastava, Hari M., Junesang Choi, "Series Associated with the Zeta and Related Functions" that contains many series. Springer 2001. But is is at the Faculty at this time of coronavirus and it is only what I have in mind, $\endgroup$
    – juan
    May 24, 2020 at 16:08
  • $\begingroup$ @juan Thank you for your suggestion. I've checked out the table of contents and some of the other pages, and it does seem to include a number of rational zeta series $\endgroup$
    – Max Muller
    May 24, 2020 at 19:09

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A book that features a vast amount of rational zeta series is "Zeta and q-Zeta Functions and Associated Series and Integrals" by H. M. Srivastava and J. Choi (2012). It is a revised and expanded version of the book suggested by juan in the comments.

It includes the following series:

$$\sum_{k=1}^{\infty} \frac{\zeta(2k)-1}{k\cdot2^{2k}} = \log\Big{(} \frac{3 \pi}{8} \Big{)}$$

$$\sum_{k=2}^{\infty} (-1)^{k} \frac{\zeta(k)-1}{3^{k}} = \frac{11}{12} - \frac{\pi}{18}\sqrt{3} - \frac{1}{2}\log(3) $$

$$\sum_{k=1}^{\infty} \{ \zeta(k)-1 \} \Big{(} \frac{4}{3} \Big{)}^{2k} = \frac{39}{14} - \frac{2}{9}\sqrt{3} $$ $$\sum_{k=1}^{\infty} \frac{2^{2k+1}-1}{5^{2k}}\zeta(2k+1) = \frac{5}{4}\log(5) - \frac{\sqrt{5}}{2}\log(2+\sqrt{5})$$

and a great deal more.

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