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Is there a known function of the form:

$$\zeta(s,a) = \displaystyle\sum_{n=1}^\infty\frac{1}{n^s+a},$$

and if so what is its name?

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    I don't know if your function has a name, but it is similar to the Hurwitz zeta function $\displaystyle \zeta(s,a) = \sum_{n = 1}^\infty \frac{1}{(n+a)^s}$.2012-08-02
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    @HenryT.Horton I am not able to discern the claimed resemblance.2012-08-03

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Let's assume $a$ is small, i.e. $0Hurwitz zeta-function.