I'm wondering if anyone knows if this function comes up anywhere in mathematics:
$$f(s)=\sum_{n=0}^\infty\frac{(-1)^n}{n!}\zeta(-ns)$$
where $\zeta(s)$ is the Riemann Zeta function.
I'm asking because it appeared for me as a term in an analytic continuation that I was doing. So I thought about just looking at this piece and trying to understand its properties.
The series I was working on continuing is:
$$ \Phi(s)=\sum_{n=1}^\infty e^{{-n}^s} $$
and using the key fact:
$$\sum_{n=1}^\infty (e^{{-n}^s}-1)= f(s) $$
The analytic continuation valid for $\Re(s)<1$ is:
$$\Phi(s)=\Gamma\bigg(1+\frac{1}{s}\bigg)+\sum_{n=0}^\infty\frac{(-1)^n}{n!}\zeta(-ns)$$
with poles at $s=-\frac{1}{n}$ a non-isolated singularity at $s=0.$ The fact that the singularities of this function accumulate at a finite point may make working with this function awkward.
Does $f(s)$ or $\Phi(s)$ come up anywhere, possibly in the theory of elliptic functions?
I asked my professor of geometry and another of analytic number theory and the geometry professor said he "knows about the last function $\Phi(s)$ but it's a long story." The analytic number theorist had not seen this function before.
I'm wondering if anyone else knows about this function and possible applications of it.
