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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 representationof the zeta function to first construct a UMM for each prime $p$. We are ableto use its phase space description to write the partition function as the traceof 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 thedata from all primes to propose a Hamiltonian and a matrix model for theRiemann zeta function.

Matrix Model for Riemann Zeta via its Local Factors