The functional equation for the Riemann zeta function is based on analysis of asymptotic behaviour for t≈0 of expression like Tr(exp(-tD^2)), where D is, say, an elliptic operator on a smooth closed manifold M. In particular, it depends heavily on the the fact that the expressions like Tr(exp(-tD^2)) have Melin transform which is holomorphic on a subspace of the complex plane of the form Re(z)>C, which is a consequence of finite dimensionality of M. We will construct an analogue of the meromorphic extension of the Riemann zeta function and prove the corresponding functional equation in the infinite dimensional limit case.

We will sketch some work in progress which give applications of these constructions to local index formulas for operators associated to infinite dimensional physical systems.