Definition of $a=0$ limit of Hurwitz Zeta function

Question

The definition of Hurwitz zeta function is $$ \zeta(s,a) =\sum_{k=0}^\infty \frac{1}{(k+a)^s} $$ where $a=0$ limit is obvious singular. But in functionsite: https://functions.wolfram.com/ZetaFunctionsandPolylogarithms/Zeta2/03/01/01/01/ $$ \zeta(s,0) \equiv \zeta(s) $$ What is the underlying algorithm to get rid of the divergence piece?

I am asking this but not looking for some convention definition. Instead in physics computation, zero mode of certain Cartan generators will lead to this kind of divergence when one does the zeta function regularization.

Answer 1

I shall not repeat @Gary's comment.

What you can write is $$\zeta (s,a)=a^{-s}+\sum_{n=0}^\infty (-1)^n\left(\frac{\zeta(s+n)}{n!} \,\prod_{k=0}^{n-1}(s+k)\right)\,a^n$$ which is exact as long as $a\neq 0$.