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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2$\begingroup$ I don't know if your function has a name, but it is similar to the Hurwitz zeta function $\displaystyle \zeta(s,a) = \sum_{n = 1}^\infty \frac{1}{(n+a)^s}$. $\endgroup$– Henry T. HortonCommented Aug 2, 2012 at 22:47
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1$\begingroup$ @HenryT.Horton I am not able to discern the claimed resemblance. $\endgroup$– SashaCommented Aug 3, 2012 at 19:56
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1 Answer
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Let's assume $a$ is small, i.e. $0<a<1$. Then $$ \sum_{n=1}^\infty \frac{1}{n^s + a} = \sum_{n=1}^\infty \frac{1}{n^s} \frac{1}{1 + a n^{-s}} = \sum_{m=1}^\infty (-a)^{m-1} \zeta(m s) $$ If $a$ is larger in absolute value, we may need to keep $p$ initial terms, so as to make sure that $|a| < (p+1)^{s}$: $$ \sum_{n=1}^\infty \frac{1}{n^s + a} = \sum_{n=1}^{p} \frac{1}{n^s + a} + \sum_{m=1}^\infty (-a)^{m-1} \zeta(m s, p+1) $$ where $\zeta(s,a)$ stands for the Hurwitz zeta-function.
