  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.
