Quantum mechanical potentials related to the prime numbers and Riemann zeros

TL;DR

This paper explores the connection between prime numbers, Riemann zeros, and quantum potentials, constructing specific potentials with eigenvalues related to these mathematical entities and analyzing their multifractal properties.

Contribution

It introduces a method to construct quantum potentials with energy levels corresponding to primes and Riemann zeros, revealing their multifractal nature.

Findings

Potentials with eigenvalues at prime numbers and Riemann zeros were successfully constructed.

The potentials exhibit multifractal characteristics as shown by R{é}nyi dimension analysis.

Results suggest new avenues for understanding the link between quantum systems and number theory.

Abstract

Prime numbers are the building blocks of our arithmetic, however, their distribution still poses fundamental questions. Bernhard Riemann showed that the distribution of primes could be given explicitly if one knew the distribution of the non-trivial zeros of the Riemann $\zeta(s)$ function. According to the Hilbert-P{\'o}lya conjecture there exists a Hermitean operator of which the eigenvalues coincide with the real part of the non-trivial zeros of $\zeta(s)$. This idea encourages physicists to examine the properties of such possible operators, and they have found interesting connections between the distribution of zeros and the distribution of energy eigenvalues of quantum systems. We apply the Mar{\v{c}}henko approach to construct potentials with energy eigenvalues equal to the prime numbers and to the zeros of the $\zeta(s)$ function. We demonstrate the multifractal nature of these potentials by measuring the R{\'e}nyi dimension of their graphs. Our results offer hope for further analytical progress.