About 6 months ago I asked for an analytic continuation of $\varphi(s)=\sum_{n\ge1} e^{-n^s}.$
From the answers and feedback I got, we know that an analytic continuation exists for $0<s<1$ and there is no analytic continuation for $s>1.$
What's the maximal analytic continuation of $\varphi(s)?$
Doing this will help me better understand how the function behaves.
As is stated in the comments, the main question is whether the line $\Re z=1$ is the natural boundary for the analytic continuation:
$$ \varphi(s)=\Gamma\left(1+\frac1s\right)+\sum_{n=0}^\infty\frac{(-1)^n}{n!}\zeta(-ns).$$
As noted by metamorphy, this series converges for complex $s\ne0$ with $\Re s<1$.