Extended Fractal Fits to Riemann Zeros

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.