Quantum graphs whose spectra mimic the zeros of the Riemann zeta function

TL;DR

This paper constructs quantum graphs whose spectral properties replicate the oscillating part of the density of states of the Riemann zeros, providing insights into the Riemann hypothesis through quantum chaos analogy.

Contribution

It introduces a novel set of quantum graphs that mimic the oscillating density of Riemann zeros, advancing the quantum chaos approach to understanding the hypothesis.

Findings

Quantum graphs replicate the oscillating part of the Riemann zeros' density of states.

The constructed graphs explain the negative sign in the oscillating part.

The smooth part of the spectrum remains different from the Riemann zeros.

Abstract

One of the most famous problems in mathematics is the Riemann hypothesis: that the non-trivial zeros of the Riemann zeta function lie on a line in the complex plane. One way to prove the hypothesis would be to identify the zeros as eigenvalues of a Hermitian operator, many of whose properties can be derived through the analogy to quantum chaos. Using this, we construct a set of quantum graphs that have the same oscillating part of the density of states as the Riemann zeros, offering an explanation of the overall minus sign. The smooth part is completely different, and hence also the spectrum, but the graphs pick out the low-lying zeros.