  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 -- and similarly 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,g). Setting d=3/2,
we estimate the frequency parameter (gamma), plus an overall scaling parameter
(sigma) we introduce. We search for that pair of parameters (gamma,sigma) which
minimizes the least-squares fit S_{n}(gamma,sigma) of the lowest n eigenvalues
-- obtained by solving the one-dimensional stationary (non-fractal) Schrodinger
equation with the trial potential (smooth plus nonsmooth parts) -- to the
lowest n 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 gotten by
using just the smooth part of the Wu-Sprung potential without any fractal
supplementation. Some limited improvement -- 5.7261 vs. 6.39207 (n=25), 11.2672
vs. 11.7002 (n=50) and 16.3119 vs. 16.6809 (n=75) -- is found in our
(non-optimized, 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.
