Consider a map $\psi:\Bbb R^2\to \Bbb R^2_+$ with $\psi(x,y)=(e^x,e^y).$ Looking at $\varphi(x)=e^{\frac{1}{\log(x)}}$ we see that it's related to $h(x)=\frac{1}{x}$ via $u=\log(x)$ and $v=\log(\varphi),$ yielding $uv=1.$
I'll try to describe a process:
$(1)$ Change the coordinate system of the plane with $\psi$.
$(2)$ Use a substituition $n\mapsto n^s.$
$(3)$ Sum the resultant function in the new coordinate system.
$(4)$ Find the analytic continuation of the resultant function.
Here's what this looks like in an example:
I started with $h(x)=\frac{1}{x},$ changed the coordinate system and got the new related function $\varphi(x).$ Next I did $n\mapsto n^s$ and then I summed, $\Phi(s)=\sum_{n=1}^\infty e^{-n^s}$ and finally asked about its analytic continuation.
With the help of user @metamorphy an analytic continuation was found
$$\Phi(s)=\Gamma\left(1+\frac1s\right)+\sum_{n=0}^\infty\frac{(-1)^n}{n!}\zeta(-ns).$$See this question.
Is there a simpler example (a cleaner analytic continuation) someone can share using this process?
Is there something I can Google to read more about such a process?
