Abstract
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.
Highlights
The zeroes of the Riemann zeta function on the critical line do not appear to fall into any pattern, but rather seem to be randomly distributed
This helped in the construction of a one-plaquette unitary random matrices (UMM) starting with the non-trivial zeroes of the Riemann zeta function, which further gave a density function in the phase space [16]
The partition function of the unitary matrix model can be expressed as an integral over a phase space, in which the angular coordinates θ are augmented by theircoDnjθugDahteem−Hom(θ,ehn),tawhh.eTreheHobisjeactiHveamisilttoonwiarnit.eWthee partition function as Z ∼ shall review this phase space description, developed in Refs. [16,17] with the motivation that phase space picture may suggest a natural
Summary
The zeroes of the Riemann zeta function on the critical line do not appear to fall into any pattern, but rather seem to be randomly distributed (see section 2 for details). Given an infinite set of distinguished points on a curve, e.g., the non-trivial zeroes of the Riemann zeta on the critical line, one can construct a random ensemble of unitary matrices (UMM) after conformally mapping the points on the unit circle. The motivation for this work is the Montgomery-Dyson observation relating the statistical distribution of the eigenvalues of an RMM and that of the nontrivial Riemann zeroes, augmented by recent advancement in our understanding of the phase space underlying an RMM This helped in the construction of a one-plaquette UMM starting with the non-trivial zeroes of the Riemann zeta function, which further gave a density function in the phase space [16].
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