A Numerical Examination of the Castro-Mahecha Supersymmetric Model of the Riemann Zeros

Abstract

The unknown parameters of the recently-proposed (Int J. Geom. Meth. Mod. Phys. 1, 751 [2004]) Castro-Mahecha model of the imaginary parts (lambda_{j}) of the nontrivial Riemann zeros are the phases (alpha_{k}) and the frequency parameter (gamma) of the Weierstrass function of fractal dimension D=3/2 and the turning points (x_{j}) of the supersymmetric potential-squared Phi^2(x) -- which incorporates the smooth Wu-Sprung potential (Phys. Rev. E 48, 2595 [1993]), giving the average level density of the Riemann zeros. We conduct numerical investigations to estimate/determine these parameters -- as well as a parameter (sigma) we introduce to scale the fractal contribution. Our primary analyses involve two sets of coupled equations: one set being of the form Phi^{2}(x_{j}) = lambda_{j}, and the other set corresponding to the fractal extension -- according to an ansatz of Castro and Mahecha -- of the Comtet-Bandrauk-Campbell (CBC) quasi-classical quantization conditions for good supersymmetry. Our analyses suggest the possibility strongly that gamma converges to its theoretical lower bound of 1, and the possibility that all the phases (alpha_{k}) should be set to zero. We also uncover interesting formulas for certain fractal turning points.