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

Dominik Śliwiński

TL;DR

This work analyzes the spectral behavior of the Connes-Consani-Moscovici family $D_{\log}^{(\lambda, N)}$, aiming to align its spectrum with the nontrivial zeros of $\zeta(\tfrac{1}{2}+is)$. It introduces the Hilbert space $\mathcal{H}_{\lambda}$, the compressed scaling generator $D_{\log}^{(\lambda,N)}$, and two error metrics $\epsilon(\lambda,N)$ and $\mathcal{E}(\lambda,N)$ to quantify spectral proximity. A central result proves an inverse-logarithmic lower bound $\epsilon(\lambda,N) \ge \tfrac{1}{4\ln\lambda}$, leading to $\epsilon(N)=\Omega(1/\ln N)$ under a natural $N\ln\lambda$ scaling; numerical results further suggest a similar inverse-log behavior for $\mathcal{E}(\lambda,N)$. The authors conjecture the existence of a limit for $\mathcal{E}(\kappa)\ln\kappa$ (potentially nonzero), and discuss the striking implication that if this limit vanishes, the spectrum would converge to $\zeta$ zeros, effectively connecting spectral data to the Riemann Hypothesis and prime distribution through the analyzed operator framework.

Abstract

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

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

TL;DR

This work analyzes the spectral behavior of the Connes-Consani-Moscovici family , aiming to align its spectrum with the nontrivial zeros of . It introduces the Hilbert space , the compressed scaling generator , and two error metrics and to quantify spectral proximity. A central result proves an inverse-logarithmic lower bound , leading to under a natural scaling; numerical results further suggest a similar inverse-log behavior for . The authors conjecture the existence of a limit for (potentially nonzero), and discuss the striking implication that if this limit vanishes, the spectrum would converge to zeros, effectively connecting spectral data to the Riemann Hypothesis and prime distribution through the analyzed operator framework.

Abstract

This paper investigates the recent Connes-Consani-Moscovici operators, whose spectra are currently hypothesized to approach the zeros of as . It turns out that when considering different standard notions of error, the dissonance between the spectra and Riemann zeros either appears to or can be proven to be inverse logarithmic in nature, which elegantly fits the distribution of prime numbers.
Paper Structure (6 sections, 2 theorems, 15 equations, 3 figures)

This paper contains 6 sections, 2 theorems, 15 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The $\mathcal{H}_\lambda$ space and the $D_{\log}^{(\lambda, N)}$ operator
  4. The errors $\epsilon(\lambda, N)$ and $\mathcal{E}(\lambda, N)$
  5. Inverse Logarithmic Lower Bound of $\epsilon$
  6. Numerical Evidence for a Stronger Statement

Key Result

Theorem 3.1

The error $\epsilon(\lambda, N)$ satisfies the weak lower bound inequality

Figures (3)

  • Figure 1: Minimum and maximum values of $\mathcal{E}(\kappa)$ around points consisting of multiples of $50$ from $50$ to $7500$ and an example logarithmic function showing the general nature of the error.
  • Figure 2: Visual representation of the highly matching behavior of the first zeros of $\zeta\left(\frac{1}{2}+is\right)$ and eigenvalues of $D_{\log}^{(7050, 7050)}$. The red lines are the imaginary parts of the Riemann $\zeta$ zeros and the dotted blue lines are the eigenvalues.
  • Figure 3: Comparison between the last $25$ out of $1000$ calculated $\zeta\left(\frac{1}{2} + is\right)$ zeros and the eigenvalues of $D_{\log}^{(7050, 7050)}$.

Theorems & Definitions (10)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • proof
  • Conjecture 4.1