Extended Fractal Fits to Riemann Zeros

TL;DR

This paper extends the analysis of Riemann zeros using fractal models to the first 300 zeros, finding limited but notable improvements in fitting the zeros with a fractal potential approach.

Contribution

It introduces an extended fractal modeling approach for Riemann zeros, analyzing a larger dataset and identifying parameter subdomains that improve fit over smooth models.

Findings

Limited improvement over smooth models in fitting zeros

Identification of two distinct parameter subdomains with better fits

Extension from 75 to 300 zeros in fractal analysis

Abstract

We extend to the first 300 Riemann zeros, the form of analysis reported by us in arXiv:math-ph/0606005, in which the largest study had involved the first 75 zeros. Again, we model the nonsmooth fluctuating part of the Wu-Sprung potential, which reproduces the Riemann zeros, by the alternating-sign sine series fractal of Berry and Lewis A(x,g). Setting the fractal dimension equal to 3/2. we estimate the frequency parameter (g), plus an overall scaling parameter (s) introduced. We search for that pair of parameters (g,s) which minimizes the least-squares fit of the lowest 300 eigenvalues -- obtained by solving the one-dimensional stationary (non-fractal) Schrodinger equation with the trial potential (smooth plus nonsmooth parts) -- to the first 300 Riemann zeros. We randomly sample values within the rectangle 0 < s < 3, 0 < g < 25. The fits obtained are compared to those gotten by using simply the smooth part of the Wu-Sprung potential without any fractal supplementation. Some limited improvement is again found. There are two (primary and secondary) quite distinct subdomains, in which the values giving improvements in fit are concentrated.