While thinking about re-constituting real symplectic manifolds into complex ones via tori fibrations and mirror symmetry I thought about a possible association of objects:
$g(x) \to \sum_n g(n^{-s})\to F(s)$
where the arrows mean the objects are corresponded, and $F(s)$ is the analytic extension of the series.
Consider the following special case:
Is it possible to associate $g(x)=1/x$ to a complex function (a series) which sums up it's heights above the x-axis at prescribed x-values?
So this for example, would correspond $g$ to $\zeta(s)$ where $\zeta(s)$ is the Riemann Zeta function.
Then would it be possible to take the surface of revolution (hyperbola of one sheet) and associate this surface with a collection of analytic extensions of its fibers where each fiber is associated to its corresponding series?
Here each of the fibers would correspond to a copy of $\zeta(s).$ I'm wondering if these associations are kosher.