Successive generation of nontrivial Riemann zeros from a Wu-Sprung type potential

TL;DR

This work targets the inverse spectral problem of reproducing Riemann zeros via a 1D symmetric Schrödinger operator. It constructs a smooth potential from the Riemann–von Mangoldt formula using a WKB constraint and then incrementally adds correction functions whose amplitudes scale with the smooth-approximation error, achieving accurate recovery of the first several zeros. The analysis links local oscillation frequencies to semiclassical momentum and demonstrates that a 50-step reconstruction recovers the first 50 zeros, with fractal patterns emerging in the potential. These results highlight that the dominant driver of the inverse problem is the smooth approximation error, suggesting a potential analytic bridge toward insights on the Riemann hypothesis.

Abstract

A series of numerical experiments are performed, where a symmetric potential is generated for the 1D time-independent Schrödinger equation, with an eigenspectrum that matches the imaginary part of the first nontrivial zeros of the Riemann Zeta Function. The potential is generated as a series of correction functions, where the starting point is a potential that matches the smooth Riemann -- von Mangoldt approximation. It is found that the correction functions display a clear pattern that can be explained in simple terms, almost entirely dependent on the approximation error in the Riemann -- von Mangoldt formula. This also provides an explanation for the fractal pattern in the potential that was observed by Wu and Sprung.

Figures (21)

• Figure 1: The imaginary part of the first 50 nontrivial Riemann zeros and the Riemann - von Mangoldt smooth approximation.
• Figure 2: Smooth potential with eigenvalues matching the approximate location of the nontrivial Riemann zeros, calculated from the Riemann - von Mangoldt formula. The figure on the right provides a zoomed-in view.
• Figure 3: Eigenvalue spectrum of the potentials in FIG. \ref{['potentials_1_4']}.
• Figure 4: Successive corrections to the smooth potential $V_0$ in FIG. \ref{['potential0']} for matching the first four nontrivial Riemann zeros.
• Figure 5: Successive corrected potentials for matching the first four nontrivial Riemann zeros.