Matrix Model for Riemann Zeta via its Local Factors

We propose the construction of an ensemble of unitary random matrices (UMM) for the Riemann zeta function. Our approach to this problem is `$p$-iecemeal', in the sense that we consider each factor in the Euler product representation of the zeta function to first construct a UMM for each prime $p$. We are able to use its phase space description to write the partition function as the trace of an operator that acts on a subspace of square-integrable functions on the $p$-adic field. This suggests a Berry-Keating type Hamiltonian. We combine the data from all primes to propose a Hamiltonian and a matrix model for the Riemann zeta function.