THE RIEMANN ZETA-FUNCTION AND THE INVERTED HARMONIC-OSCILLATOR

The Riemann zeta function has phase jumps of pi every time it changes sign as the parameter t in the complex argument s = 1/2 + it is varied . We show analytically that as the real part of the argument is increa sed to sigma > 1/2, the memory of the zeros fades only gradually throu gh a Lorentzian smoothing of the density of the zeros. The correspondi ng trace formula, for sigma much greater than 1, is of the same form a s that generated by a one-dimensional harmonic oscillator in one direc tion, along with an inverted oscillator in the transverse direction. I t is pointed out that Lorentzian smoothing of the level density for mo re general dynamical systems may be done similarly. The Gutzwiller tra ce formula for the simple saddle plus oscillator model is obtained ana lytically, and is found to agree with the quantum result. (C) 1997 Aca demic Press.