About 6 months ago I [asked][1] 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 $s>1.$

>I'd like to understand whether or not there is an analytic continuation for $s<0.$


  [1]: https://mathoverflow.net/questions/368647/what-is-the-analytic-continuation-of-varphis-sum-n-ge-1-e-ns

Doing this will help me better understand how the function behaves.