In equation $6.6$ pg. $20$ of https://www.researchgate.net/publication/318529311_Some_infinite_series_involving_hyperbolic_functions we have a nice expression $$\sum_{n=1}^{\infty}\frac{1}{n(e^{2\pi n}-1)}=\frac{1}{4}\log\left(\frac{4}{\pi}\right)-\frac{\pi}{12}+\log \Gamma \left(\frac{3}{4}\right) $$ Question: I have searched the above referenced book but could not find a proof of the above formula. Can someone please prove it or give a reference (with page number) of it?
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1Have you searched the formula on approach0? Related question. – nejimban Feb 17 '23 at 05:27
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@nejimban Thank you. But for a proof, Can you please give a reference of a book? Or a proof please? – Max Feb 17 '23 at 05:30
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The second link refers to “Ramanujan's Notebooks Volume 2 by B.C. Berndt”. The proof for the formula apparently uses the theory of elliptic functions, of which I am not familiar. I do not know of any other proof, but hopefully the first link (approach0) points at other references. – nejimban Feb 17 '23 at 05:32
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@nejimban Okay. Thanks a lot – Max Feb 17 '23 at 05:49