The Weyl calculus and the zeta function (Q931796)

Let \(\Gamma=\text{SL}_2({\mathbb Z})\). Given \(N\), which is \(4\) times a square-free odd positive integer, the author defines a family \((\omega_\rho)_\rho\) of tempered distributions on the line, which are linear combinations of combs and are parametrized by the set \(R\) of squares \(\rho\) in the group \(({\mathbb Z}/N{\mathbb Z})^\times\). The family has the property that its linear span is stable under the action of the metaplectic cover of \(\Gamma\). Let \(W_N\) be the Weyl symbol of the operator \(u\mapsto\sum_{\rho\in R}\langle \omega_\rho,u\rangle\omega_\rho\). The following result is presented in the paper: One has \[ W_N=N^{\mathcal E}\zeta_N({\mathcal E})^{-1}D_0, \] where \(\mathcal E\) is the operator \(x\frac{\partial}{\partial x}+\xi\frac{\partial}{\partial\xi}+1\) on the \((x,\xi)\)-plane, \(D_0\) is the standard Dirac-comb and \[ \zeta_N(s)\;=\;\prod_{p|N}(1-p^{-s})^{-1}, \] where the product is extended over all primes \(p\) dividing \(N\). The latter is a finite cutout of the product expansion of the Riemann zeta function, whence the title.