Riemann Hypothesis: Architecture of a conjecture "along" the lines of P\'olya. From trivial zeros and Harmonic Oscillator to information about non-trivial zeros of the Riemann zeta-function

TL;DR

This paper proposes a novel Hamiltonian-based framework linking the Riemann Hypothesis to the spectrum of a constructed operator that matches the harmonic oscillator if and only if the hypothesis is true.

Contribution

It introduces a new conjectural architecture involving a Hamiltonian operator whose spectrum encodes the truth of the Riemann Hypothesis, offering an alternative to Pólya's strategy.

Findings

Constructed Hamiltonian spectrum matches harmonic oscillator if RH holds

Formulated a non-commutative structure on the real axis related to RH

Provided a new operator equation characterizing the Riemann Hypothesis

Abstract

We propose an architecture of a conjecture concerning the Riemann Hypothesis in the form of an "alternative" to the P\'olya strategy: we construct a Hamiltonian H_Polya whose spectrum coincides exactly with that of the Harmonic Oscillator Hamiltonian H_osc if and only if the Riemann Hypothesis holds true. In other words, it can be said that we formulate the Riemann Hypothesis by means of a non-commutative structure on the real axis, viz., that of the Harmonic Oscillator, by an equation of the type H_Polya(H_osc) = H_osc: the Harmonic Oscillator operator, if viewed as an argument of H_Polya, reproduces itself.