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# An series related to the eta Dedekind function

Prove that

Solution

Let us consider the function

and integrate it along a quadratic counterclockwise contour with vertices where is a big odd natural number. Hence,

We note that and that .

It’s also easy to see that

Hence, as we have that

By Residue theorem we have that

It is straightforward to show that

Hence,

in the limit . The result follows.