Quantum simulation of the Riemann-Hurwitz ζ function

We propose a simple realization of a quantum simulator of the Riemann-Hurwitz (RH) ζ function based on a truncation of its Dirichlet representation. We synthesize a nearest-neighbor-interaction Hamiltonian, satisfying the property that the temporal evolution of the autocorrelation function of an initial bare state of the Hamiltonian reproduces the RH function along the line σ + iωt of the complex plane, with σ > 1. The tight-binding Hamiltonian with engineered hopping rates and site energies can be implemented in a variety of physical systems, including trapped-ion systems and optical waveguide arrays. The proposed method is scalable, which means that the simulation can be, in principle, arbitrarily accurate. Practical limitations of the suggested scheme, arising from a finite number of lattice sites N and from decoherence, are briefly discussed.