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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    $\begingroup$ if $ c\to 0 $ and you assume we can do renormalization, we can neglect the secon term to get $ \zeta_{M}(s,c)=c^{2s}\zeta(2s) $ so the function can be expressed in terms of the Riemann zeta function. In case $ c\to \infty$ by the same reasoning we can neglect the first term so $ \zeta_{M}c^{-2s}\Zeta (-2s) $ in fact you can see that your function is invariant under scalling of the 'c' so $\Zeta_{M}(s,c)=\zeta_{M}(s,1/c)$ $\endgroup$ – Jose Garcia Nov 16 '11 at 14:55
  • $\begingroup$ $\zeta_M(s,c) = \sum c^{2s} n^{-2s} (1 + (c/n)^4)^{-s}$, and $(1+x)^{-s} = \frac{1}{\Gamma(s)} \sum_{k=0}^\infty x^{k} (-1)^k \frac{\textstyle\Gamma(s+k)}{\textstyle k! }$ $\endgroup$ – reuns Jul 19 '15 at 0:20

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