On functions similar to Hurwitz zeta function

Question

Denoted as $\zeta(s,a)$ for a > 0

Where do I find topics on the Hurwitz zeta function for a < 0?

Any links or resources would be appreciated. (Please dont mention wiki or mathworld)

Thanks

Answer 1

Well, there's the DLMF and the Wolfram Functions site... which people really should be checking out first when they encounter an unfamiliar "special function".

Answer 2

There is a very simple connection between $\zeta(s,a)$ when $a$ is negative vs when $a$ is positive.

First you need to know that $\zeta(s,a)$ is undefined when $a$ is a negative integer. This is because then $(n+a)^{-1}$ can be undefined when $n=-a.$ So you must let $a$ be not an integer.

If you do that, we can then use an identity: $\zeta(s,a)= \frac{1}{a} + \zeta(s,a+1)$ which holds by analytic continuation to $\mathbb{C}\setminus{1}.$ By using this repeatedly, we can eventually shift the zeta function to a positive value.