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On the Hilbert space derived from the Weil distribution

Masatoshi Suzuki

TL;DR

The paper investigates the Hilbert space $\mathcal{H}_W$ obtained by completing $C_c^\infty(\mathbb{R})$ with respect to the Weil hermitian form arising from the Weil distribution, under the RH. It proves that, assuming RH, $\mathcal{H}_W$ is isomorphic to a de Branges space $\mathcal{H}(E_\xi)$ and to a model space $\mathcal{K}(\Theta_\xi)$ via a Fourier-based isomorphism, tying the arithmetic data encoded by the zeros of $\xi(1/2-iz)$ to a concrete Hilbert-space framework. The authors introduce unconditional constructions $\mathcal{H}_0$ and $\mathcal{K}_0$, which are shown to coincide with $\mathcal{H}_W$ and $\mathcal{K}(\Theta_\xi)$ under RH, yielding an equivalence condition for RH expressed as explicit norm equalities on a screw-line parametrization. They also formulate a Hilbert–Pólya interpretation of $\mathcal{H}_W$, showing that, under RH, the zeros $\Gamma$ appear as eigenvalues of a self-adjoint operator, thereby linking the spectral perspective with the explicit formula. Overall, the work translates RH into equalities between $L^2$-norms in a de Branges/model-space setting and provides a robust, structure-rich platform for further RH investigations.

Abstract

We study the Hilbert space obtained by completing the space of all smooth and compactly supported functions on the real line with respect to the hermitian form arising from the Weil distribution under the Riemann hypothesis. It turns out that this Hilbert space is isomorphic to a de Branges space by a composition of the Fourier transform and a simple map.This result is applied to state a new equivalence condition for the Riemann hypothesis in a series of equalities.

On the Hilbert space derived from the Weil distribution

TL;DR

The paper investigates the Hilbert space obtained by completing with respect to the Weil hermitian form arising from the Weil distribution, under the RH. It proves that, assuming RH, is isomorphic to a de Branges space and to a model space via a Fourier-based isomorphism, tying the arithmetic data encoded by the zeros of to a concrete Hilbert-space framework. The authors introduce unconditional constructions and , which are shown to coincide with and under RH, yielding an equivalence condition for RH expressed as explicit norm equalities on a screw-line parametrization. They also formulate a Hilbert–Pólya interpretation of , showing that, under RH, the zeros appear as eigenvalues of a self-adjoint operator, thereby linking the spectral perspective with the explicit formula. Overall, the work translates RH into equalities between -norms in a de Branges/model-space setting and provides a robust, structure-rich platform for further RH investigations.

Abstract

We study the Hilbert space obtained by completing the space of all smooth and compactly supported functions on the real line with respect to the hermitian form arising from the Weil distribution under the Riemann hypothesis. It turns out that this Hilbert space is isomorphic to a de Branges space by a composition of the Fourier transform and a simple map.This result is applied to state a new equivalence condition for the Riemann hypothesis in a series of equalities.
Paper Structure (18 sections, 21 theorems, 96 equations)

This paper contains 18 sections, 21 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Review on necessary notions
  3. Screw functions and screw lines
  4. Hilbert spaces associated with screw functions
  5. De Branges spaces
  6. Model subspaces
  7. Unconditional results
  8. Expansion of $\mathfrak{P}_t(z)$ over the zeros
  9. Proof of Proposition \ref{['prop_101']}
  10. Two special Hilbert spaces
  11. A screw line of the Riemann zeta-function
  12. A special orthonormal basis
  13. Screw line of the Riemann zeta-function
  14. Proof of Theorem \ref{['thm_1_2']}
  15. Proof of Theorem \ref{['thm_1_1']} and its refinement
  16. ...and 3 more sections

Key Result

Theorem 1.1

Assume that the RH holds. Let $\mathcal{H}_W$, $\mathcal{H}(E_\xi)$, and $\mathcal{K}(\Theta_\xi)$ be Hilbert spaces as above. Then, the map $\mathcal{K}(\Theta_\xi) \to \mathcal{H}_W$ defined by is an isomorphism between Hilbert spaces and satisfies for $F \in \mathcal{K}(\Theta_\xi)$, where $\mathsf{F}^{-1}$ is the Fourier inversion on $L^2(\mathbb{R})$.

Theorems & Definitions (41)

  • Theorem 1.1
  • Proposition 1.2
  • proof
  • Proposition 1.3
  • Theorem 1.4
  • proof
  • Corollary 1.5
  • proof
  • Proposition 3.1
  • proof
  • ...and 31 more