I'm looking for a sequence $f(n)$, so that
$g(z):=\lim_{N\to\infty}\sum_{n=0}^N\exp\left(-z\cdot f(n)\right),$
with $z$ so that this converges classically, defines a function which can not be analytically continued, i.e. the sum is as good as it gets.
For a non-example, plugging using $f(n)=n$ leads, for $z>0$ and ${\mathrm{e}}^{-z}<1$, to $\frac{1}{1-{\mathrm{e}}^{-z}}$, but this can be extended to make sense for $z=-9001$. Plugging in $f(n)=\log(n)$ leads to the zeta function. All examples where I'm able to compute the sum for some $z$ lead to something extendable.
PS: It would also be great if the result is ${\mathcal C}^\infty$.