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I am interested in the following generalization of the Riemann Zeta function: $ \zeta_M(s,c) = \sum_{n=1}^\infty \left(\frac{n^2}{c^2} + \frac{c^2}{n^2}\right)^{-s} $

This is most closely related (in spirit) to the Epstein-Hurwitz Zeta function $ \zeta_{EH}(s,c) = \sum_{n=1}^\infty \left(n^2+c^2\right)^{-s} $ about which there exists some interesting literature (e.g. Elizalde).

Has anyone seen this function $\zeta_M(s,c)$? Can anyone prove any interesting identities, integral representations, asymptotic expansions, relations to other zeta functions, or other useful facts about it? I am particularly interested in the symmetric limits $c\to0$, $c\to\infty$ and $s\to 1/2$ for a problem in physics. The last limit corresponds to the Zeta function pole $\zeta(1)$.

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    $\zeta_M(s,c) = \sum c^{2s} n^{-2s} (1 + (c/n)^4)^{-s}$, and $(1+x)^{-s} = \frac{1}{\Gamma(s)} \sum_{k=0}^\infty x^{k} (-1)^k \frac{\textstyle\Gamma(s+k)}{\textstyle k! }$2015-07-19

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