The Negative Prolate Spectrum and the Riemann–Siegel Theta Function

Connes and Moscovici found that the negative eigenvalues of the self-adjoint prolate wave operator reproduce the growth of the zeros of the Riemann zeta function. In the published version, their counting law for the associated Dirac operator contains a term −(log E)/2π. This paper proves that this term is absent. The proof is an elementary phase analysis, and it places the eigenvalues, up to O(t^{-2/3}), at the points where the Riemann–Siegel theta function equals (n − 3/8)π. LaTeX source, figure, PDF and a numerical check.