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

TL;DR

This paper numerically investigates the Castro-Mahecha supersymmetric model of the Riemann zeros, estimating key parameters and exploring their convergence, with implications for understanding the zeros' distribution.

Contribution

It provides the first detailed numerical analysis of the Castro-Mahecha model, estimating parameters and examining their behavior to support the model's validity.

Findings

Gamma likely converges to 1

All phases alpha_k should be zero

Derived formulas for fractal turning points

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.