Skip to main content
eScholarship
Open Access Publications from the University of California

UC Santa Barbara

UC Santa Barbara Previously Published Works bannerUC Santa Barbara

Fractal fits to Riemann zeros

Abstract

Wu and Sprung (Phys. Rev. E, 48, 2595 (1993)) reproduced the first 500 nontrivial Riemann zeros, using a one-dimensional local potential model. They concluded - as did van Zyl and Hutchinson (Phys. Rev. E, 67, 066211 (2003)) - that the potential possesses a fractal structure of dimension d = 3/2. We model the nonsmooth fluctuating part of the potential by the alternating-sign sine series fractal of Berry and Lewis A(x, gamma). Setting d = 3/2, we estimate the frequency parameter (gamma), plus an overall scaling parameter (sigma) that we introduce. We search for that pair of parameters (gamma, sigma) that minimizes the least-squares fit S-n(gamma, sigma) of the lowest n eigenvalues - obtained by solving the one-dimensional stationary (nonfractal) Schrodinger equation with the trial potential (smooth plus nonsmooth parts) - to the lowest if Riemann zeros for n = 25. For the additional cases, we study, n = 50 and 75, we simply set sigma = 1. The fits obtained are compared to those found by using just the smooth part of the Wu-Sprung potential without any fractal supplementation. Some limited improvement - 5.7261 versus 6.39207 (n = 25), 11.2672 versus 11.7002 (n = 50), and 16.3119 versus 16.6809 (n = 75) - is found in our (nonoptimized, computationally bound) search procedures. The improvements are relatively strong in the vicinities of gamma = 3 and (its square) 9. Further, we extend the Wu-Sprung semiclassical framework to include higher order corrections from the Riemann-von Mangoldt formula (beyond the leading, dominant term) into the smooth potential.

Many UC-authored scholarly publications are freely available on this site because of the UC's open access policies. Let us know how this access is important for you.

Main Content
For improved accessibility of PDF content, download the file to your device.
Current View