The Riemann zeta function and the inverted harmonic oscillator (Q1269391)

As a function of \(t\) and along the line \(\sigma=1/2\), the Riemann zeta function \(\zeta(\sigma+it),\sigma,t \in\mathbb{R}\) experiences a discontinuous jump by \(\pi\) in its phase angle, every time that it changes sign. Otherwise the phase angle is a smooth function of \(t\). In a previous paper with some common authors [\textit{R. K. Bhaduri}, \textit{A. Khare}, and \textit{J. Law}, Phys. Rev. E 52, 486 ff. (1995)], it was observed that when \(\sigma\) is increased, the phase of the zeta function becomes more and more smooth, and that the smooth part is directly linked to the quantum scattering phase shift of a one-dimensional inverted harmonic oscillator. In the present paper, the authors examine the derivative of the phase of the zeta function with respect to \(t\), for a fixed \(\sigma>1/2\), and prove that it is the Lorentz-smoothed oscillating part of the density of the zeros at \(\sigma=1/2\), which can be expressed as a Gutzwiller-like trace formula with primitive orbits whose periods are the logarithms of the prime numbers. By choosing a large \(\sigma\), the contributions of the large primes are severely damped. To investigate this behavior in a closer way, the authors then introduce a dynamical model that mimicks the Riemann zeta function for very large \(\sigma\). They see that the corresponding trace formula 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. They obtain analytically the Gutzwiller trace formula for the simple saddle plus oscillator model and find that it agrees with the corresponding quantum result. It is also pointed out in the paper that, in a similar way, Lorentzian smoothing of the level density for more general dynamical systems may be done. The final question is whether a single Hamiltonian may in fact describe the phase of the zeta function for the entire range of \(\sigma\geq 1\).