A Chiral Adelic Dirac Operator and the Spectral Realization of the Riemann Zeros

TL;DR

This paper introduces a chiral adelic Dirac operator framework on the idèle class space that encodes the functional equation of L-functions, linking spectral properties to the zeros of the Riemann zeta function through prime-indexed deformations.

Contribution

It develops a novel adelic Dirac operator with involutive symmetry and prime-based deformations, providing a spectral interpretation of the Riemann zeros and a new approach to the Hilbert-Pólya conjecture.

Findings

Eigenvalues form paired spectral gaps related to prime deformations

Spectral shift function exhibits jump discontinuities at eigenvalues

Trace formula connects spectral data with prime orbit expansions

Abstract

This paper develops a chiral adelic operator framework in which the functional--equation symmetry of global $L$--functions is realized directly in the spectrum of a Dirac--type Hamiltonian. Working on the id\`ele class space, we place a real--place Floquet Hamiltonian into an off--diagonal chiral form to obtain a global adelic Dirac operator with an exact involutive symmetry implemented by real reflection and idelic inversion. Arithmetic information is incorporated through a prime--indexed mass deformation built from spherical Hecke operators; when the coefficient functions are even, the perturbed operator preserves the chiral symmetry and produces isolated $\pm$--paired eigenvalues inside the spectral gaps of the Floquet background. These eigenvalues appear as jump discontinuities of the Dirac spectral shift function, while a separated adelic trace formula expresses the trace as a product of a Floquet orbital factor and a prime--indexed Euler--factor--type term whose logarithmic derivatives yield a prime--orbit expansion reminiscent of the explicit formula. This structure motivates a Dirac reinterpretation of the Hilbert--P\'olya idea, identifying the nontrivial zeros of $\zeta(s)$ not with the raw spectrum of a single operator but with the spectral--shift discontinuities of a chiral adelic Dirac system under controlled prime--indexed deformations, with finite--prime truncations providing computable models that converge distributionally and enable numerical exploration of arithmetic spectral flow.