Spectra of alternating Hilbert operators

TL;DR

This paper investigates the spectra of alternating Hilbert operators, aiming to connect their spectral properties with the zeros of zeta functions, and introduces a new method for calculating these spectra to inspire further research.

Contribution

It presents a novel scheme for calculating spectra of alternating Hilbert operators, potentially linking spectral analysis to the Riemann Hypothesis and zeta function zeros.

Findings

Calculated spectra for alternating Hilbert operators.

Proposed a new method for spectral computation.

Suggested connections between spectra and zeta function zeros.

Abstract

Spectra of real alternating operators seem to be quite interesting from the view point of explaining the Riemann Hypothesis for various zeta functions. Unfortunately we have not sufficient experiments concerning this theme. Necessary works would be to supply new examples of spectra related to zeros and poles of zeta functions. A century ago Hilbert (1907) considered a kind of operators representing quadratic forms of infinitely many variables. Demonstrating the calculation of spectra for alternating Hilbert operators we hope to present a novel scheme in this paper. Authors expect this study encourages experts for further studies.