Q-Zeta and Elliptic Hall Polynomials

The fundamental property of zeta functions and L-functions is that their meromorphic continuations provide a lot of information about the corresponding objects. Complex values of $s$ occur as a technical tool, with little direct arithmetic-geometric meaning. In the refined theory, $1/n^s$ are replaced by certain $q,t,a$-series, which are invariants of Lens Spaces $L(n,1)$ directly related to Elliptic Hall
Polynomials. Superduality is one of their key features: under $q\leftrightarrow 1/t, a\mapsto a$. As $t\to 0$, they become Rogers-Ramanujan series (certain string functions and those from the Nahm Conjecture), the limit to Hall Polynomials is $q\to 0$, etc. We will begin with the $q$-deformation of Riemann's zeta: the case of $A_1$ when $t=q^{s-1/2}, a=t^2$.