The Riemann zeta function and the inverted harmonic oscillator

Abstract

The Riemann zeta function has phase jumps of π every time it changes sign as the parameter t in the complex arguments=½+itis varied. We show analytically that as the real part of the argument is increased to σ>½, the memory of the zeros fades only gradually through a Lorentzian smoothing of the density of the zeros. The corresponding trace formula, for σ»1, is of the same form as that generated by a one-dimensional harmonic oscillator in one direction, along with an inverted oscillator in the transverse direction. It is pointed out that Lorentzian smoothing of the level density for more general dynamical systems may be done similarly. The Gutzwiller trace formula for the simple saddle plus oscillator model is obtained analytically, and is found to agree with the quantum result.