recently I stumbled more than once on sums of the following form
$$ \sum_{k \in \mathbb{Z} }\frac{1}{(a|k|+b)^{1+c}} $$ where $a,b,c>0$ real numbers. One can prove that this always converges, and I suppose you can try to relate it to zeta functions.
However, I would like to estimate the sum in terms of $a$ and $b$ and $c$. More particularly, I would like to know the order of the sum as $b \longrightarrow \infty$.
I tried to use integral aproximations, but they seemed to loose a lot of informations as I take $b$ large.