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
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
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".
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.