Riemann zeta function generalization

[lokofer]

"Riemann zeta function"...generalization..

Hello my question is if we define the "generalized" Riemann zeta function:

$$\zeta(x,s,h)= \sum_{n=0}^{\infty}(x+nh)^{-s}$$

which is equal to the usual "Riemann zeta function" if we set h=1, x=0 ,then my question is if we can extend the definition to include negative values of "s" (using a functional equation or something similar)..

 

[shmoe]

$$\zeta(x,s,h)= \sum_{n=0}^{\infty}(x+nh)^{-s}=h^{-s} \sum_{n=0}^{\infty}(x/h+n)^{-s}$$

It's just a Hurwitz zeta function.

 

[matt grime]

There is a whole well documented world of things like this out there, Jose. L functions, generalized zeta functions, indeed the generalized Riemann hypothesis is known to be true for many many of the generalized zeta functions.