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Nontrivial Riemann Zeros as Spectrum

Enderalp Yakaboylu

TL;DR

The paper presents a non-symmetric Riemann operator $\hat{\mathcal{R}}$ on $L^2([0,\infty))$ whose spectrum encodes the zeros of the completed eta function, including the nontrivial zeta zeros. By constructing a positive intertwiner $\hat{W}$ (via a regularized $\hat{V}_R$ and oblique projections), the authors obtain a Hilbert–Pólya-type framework in which $\hat{W}\ge0$ forces $\Re(\rho)=1/2$ for all nontrivial zeros $\rho$, thereby linking positivity to the Riemann Hypothesis. A self-adjoint Hilbert–Pólya operator $\hat{h}=\hat{W}^{1/2}\hat{\mathcal{R}}\hat{W}^{-1/2}$ is constructed, with spectrum equal to the imaginary parts of the zeros, providing a spectral realization of RH. The framework is extended to higher-order zeros and generalized to Mellin-transformable $L$-functions, suggesting a broader operator-theoretic pathway to the Generalized Riemann Hypothesis via positivity and intertwiners.

Abstract

Define $ Υ(s) := Γ(s+1)\, (1-2^{1-s}) \, ζ(s) $, and let $ Z := \left\{γ\in \mathbb{C} \;\middle|\; Υ(γ)=0 \right\} $ denote its zero set, consisting of the nontrivial zeta zeros $ρ$, together with the zeros of $ (1-2^{1-s}) $, excluding $ s = 1 $. We introduce a non-symmetric operator \[ R \colon D(R) \subset L^2([0,\infty)) \to L^2([0,\infty)) \, , \] with spectrum \[ σ(R) = \left\{ i\left(1/2- γ\right) \;\middle|\; γ\in Z \right\} \, . \] Assuming simplicity of all nontrivial zeta zeros, we construct a positive semidefinite operator $ W $ that intertwines $ R$ and its adjoint on the corresponding spectral subspace, $ R^\dagger W = W R $. The positivity of $ W $, which represents an operator-theoretic form of (Bombieri's refinement of) Weil's positivity criterion, enforces $ \Re(ρ)=1/2 $ for all $ ρ$, in accordance with the Riemann Hypothesis. The same positivity condition naturally yields a self-adjoint operator whose spectrum coincides with the imaginary parts of the nontrivial zeta zeros. We further extend the framework to accommodate higher-order zeta zeros, should they exist, and observe that it generalizes to any Mellin-transformable $L$-function satisfying a functional equation.

Nontrivial Riemann Zeros as Spectrum

TL;DR

The paper presents a non-symmetric Riemann operator on whose spectrum encodes the zeros of the completed eta function, including the nontrivial zeta zeros. By constructing a positive intertwiner (via a regularized and oblique projections), the authors obtain a Hilbert–Pólya-type framework in which forces for all nontrivial zeros , thereby linking positivity to the Riemann Hypothesis. A self-adjoint Hilbert–Pólya operator is constructed, with spectrum equal to the imaginary parts of the zeros, providing a spectral realization of RH. The framework is extended to higher-order zeros and generalized to Mellin-transformable -functions, suggesting a broader operator-theoretic pathway to the Generalized Riemann Hypothesis via positivity and intertwiners.

Abstract

Define , and let denote its zero set, consisting of the nontrivial zeta zeros , together with the zeros of , excluding . We introduce a non-symmetric operator with spectrum Assuming simplicity of all nontrivial zeta zeros, we construct a positive semidefinite operator that intertwines and its adjoint on the corresponding spectral subspace, . The positivity of , which represents an operator-theoretic form of (Bombieri's refinement of) Weil's positivity criterion, enforces for all , in accordance with the Riemann Hypothesis. The same positivity condition naturally yields a self-adjoint operator whose spectrum coincides with the imaginary parts of the nontrivial zeta zeros. We further extend the framework to accommodate higher-order zeta zeros, should they exist, and observe that it generalizes to any Mellin-transformable -function satisfying a functional equation.
Paper Structure (11 sections, 15 theorems, 105 equations)

This paper contains 11 sections, 15 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Completed Eta Function and Its Zeros
  4. Hilbert-Space Framework and Canonical Operators
  5. Riemann Operator
  6. Similarity of Riemann Operator and Its Adjoint
  7. Oblique Projections and Compressed Operators
  8. Riemann Hypothesis and Hilbert-Pólya Operator
  9. Construction of the Hilbert-Pólya Operator
  10. Riemann Hypothesis Beyond Simple Zeros
  11. Conclusion

Key Result

Proposition 2.4

The operator $\hat{D}$ is essentially self-adjoint on the core domain $\mathcal{C}$, and hence admits a unique self-adjoint extension. We use the same symbol for this extension, whose domain is denoted by $\mathcal{D}(\hat{D})$.

Theorems & Definitions (44)

  • Definition 2.1: Zeros of $\Upsilon (s)$
  • Remark 2.2
  • Definition 2.3: Operators $\hat{D}$ and $\hat{T}$
  • Proposition 2.4: Self-adjointness of $\hat{D}$
  • proof
  • Proposition 2.5: Self-adjointness of $\hat{T}$
  • proof
  • Definition 3.1: Operator $\hat{\mathcal{R}}$
  • Theorem 3.2: Spectrum and Eigenstates of $\hat{\mathcal{R}}$
  • proof
  • ...and 34 more