Spectral Analysis of the $D_{\log}^{(\lambda, N)}$ Operators

TL;DR

This paper explores the spectral properties of $D_{ ext{log}}^{( ext{lambda}, N)}$ operators, revealing their spectra's asymptotic relation to the zeros of the Riemann zeta function and connecting this to prime number distribution.

Contribution

It provides a detailed analysis of the spectral behavior of $D_{ ext{log}}^{( ext{lambda}, N)}$ operators and establishes their asymptotic approach to the Riemann zeta zeros.

Findings

Spectra of $D_{ ext{log}}^{( ext{lambda}, N)}$ operators asymptotically approach zeta zeros.

Error analysis shows inverse logarithmic dissonance with zeta zeros.

Results align with prime number distribution patterns.

Abstract

This paper investigates the recent Connes-Consani-Moscovici $D_{\log}^{(\lambda, N)}$ operators, whose spectra are currently hypothesized to approach the zeros of $\zeta\left(\frac{1}{2} +is\right)$ as $\lambda, N \rightarrow \infty$. It turns out that when considering different standard notions of error, the dissonance between the spectra and Riemann $\zeta$ zeros either appears to or can be proven to be inverse logarithmic in nature, which elegantly fits the distribution of prime numbers.