The phase of the Riemann zeta function

Abstract

We, offer an alternative interpretation of the Riemann zeta function ζ(s) as a scattering amplitude and its nontrivial zeros as the resonances in the scattering amplitude. We also look at several different facets of the phase of the ζ function. For example, we show that the smooth part of the ζ function along the line of the zeros is related to the quantum density of states of an inverted oscillator. On the other hand, for ℜs > ½, we show that the memory of the zeros fades only gradually through a Lorentzian smoothing of the delta functions. The corresponding trace formula for ℜs » 1 is shown to be of the same form as generated by a one-dimensional harmonic oscillator in one direction along with an inverted oscillator in the transverse direction. Quite remarkably for this simple model, the Gutzwiller trace formula can be obtained analytically and is found to agree with the quantum result.