About 6 months ago I asked for an analytic continuation of $\varphi(s)=\sum_{n\ge1} e^{-n^s}.$

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 $\Re s<1$.

The function is clearly analytic for $s>0$ as a real variableNo, no, and once more no! When $s>1$, it is $C^\infty$ and even in a quasi-analytic class, but notreal analytic. $\endgroup$5more comments