Spectral Flow for the Riemann zeros

TL;DR

This paper explores spectral flow in a quantum model linked to the Riemann zeros, proposing a new criterion for the Riemann Hypothesis based on spectral flow and extending the analysis to generalized hypotheses.

Contribution

It introduces a spectral flow approach to analyze the Riemann zeros and provides a simple criterion for the Riemann Hypothesis, including extensions to generalized cases.

Findings

Spectral flow criteria relate to the validity of the Riemann Hypothesis.

Counterexample shows violation when the Euler product is absent.

Spectral flow offers a new perspective on the zeros of the zeta function.

Abstract

Recently, with Mussardo we defined a quantum mechanical problem of a single particle scattering with impurities wherein the quantized energy levels $E_n (\sigma)$ are exactly equal to the zeros of the Riemann $\zeta (s)$ where $\sigma = \Re (s)$ in the limit $\sigma \to 1/2$. The S-matrix is based on the Euler product and is unitary by construction, thus the underlying hamiltonian is hermitian and all eigenvalues must be real. Motivated by the Hilbert-P\'olya idea we study the spectral flows for $\{ E_n (\sigma) \}$. This leads to a simple criterion for the validity of the Riemann Hypothesis. The spectral flow arguments are simple enough that we present analogous results for the Generalized and Grand Riemann Hypotheses. We also illustrate our results for a counter example where the Riemann Hypothesis is violated since there is no underlying unitary S-matrix due to the lack of an Euler product.