[Paper Review] A canonical system of differential equations arising from the Riemann zeta-function

Masatoshi Suzuki|arXiv (Cornell University)|Apr 9, 2012

advanced mathematical theories20 references3 citations

TL;DR

This paper establishes a canonical system of differential equations derived from the Riemann xi-function, linking the Riemann Hypothesis (RH) to the meromorphic inner function property of Θω(z) = ξ(1/2−ω−iz)/ξ(1/2+ω−iz) for ω > 1. It constructs the system unconditionally via Fredholm determinants of compact integral operators, providing a criterion for RH through positive semidefiniteness of associated Hamiltonian matrices.

ABSTRACT

This paper has two main results, which relate to a criteria for the Riemann hypothesis via the family of functions $Θ_ω(z)=ξ(1/2-ω-iz)/ξ(1/2+ω-iz)$, where $ω>0$ is a real parameter and $ξ(s)$ is the Riemann xi-function. The first main result is necessary and sufficient conditions for $Θ_ω$ to be a meromorphic inner function in the upper half-plane. It is related to the Riemann hypothesis directly whether $Θ_ω$ is a meromorphic inner function. In comparison with this, a relation of the Riemann hypothesis and the second main result is indirect. It relates to the theory of de Branges, which associates a meromorphic inner function and a canonical system of linear differential equations (in the sense of de Branges). As the second main result, the canonical system associated with $Θ_ω$ is constructed explicitly and unconditionally under the restriction of the parameter $ω>1$ by applying a method of J.-F. Burnol in his recent work on the gamma function to the Riemann xi-function. If such construction is extended to all $ω> 0$ unconditionally, we get a criterion for the Riemann hypothesis in terms of a family of canonical systems parametrized by $ω>0$, which explains the validity of the Riemann hypothesis as positive semidefiniteness of the corresponding family of Hamiltonian matrices.

Motivation & Objective

• To establish a canonical system of linear differential equations associated with the Riemann xi-function for ω > 1, unconditionally and without assuming RH.
• To connect the Riemann Hypothesis to the meromorphic inner function property of Θω(z) in the upper half-plane.
• To provide a constructive framework for realizing the zeros of Aω(z) as eigenvalues of a positive operator via de Branges space theory.
• To extend the canonical system construction beyond ω ≥ 1/2 to ω > 1, using J.-F. Burnol’s method on the gamma function and the xi-function.
• To propose a criterion for RH based on the positive semidefiniteness of Hamiltonian matrices derived from the canonical system.

Proposed method

• Define Θω(z) = ξ(1/2−ω−iz)/ξ(1/2+ω−iz), where ξ(s) is the Riemann xi-function, and analyze its analytic properties in the upper half-plane.
• Use the functional equations of ξ(s) to derive symmetry properties of Aω(z) and Bω(z), which are real on the real line and even/odd respectively.
• Apply Burnol’s method on the gamma function to express the kernel hω⟨1⟩(x) as a Mellin-Barnes integral involving Θω(z), ensuring convergence for ω > 1.
• Construct the canonical system via Fredholm determinants of compact integral operators, linking the system to the spectral theory of de Branges spaces.
• Use the residue theorem and asymptotic analysis (via Stirling’s formula) to derive the integral representation of x−1/2 − hω⟨1⟩(x) in L²((1,∞), dx).
• Establish equivalence between the L²-integrability of x−1/2 − hω⟨1⟩(x) and the inner function property of Θω(z), leading to the canonical system construction.

Experimental results

Research questions

• RQ1 Under what conditions is Θω(z) a meromorphic inner function in the upper half-plane, and how is this related to the Riemann Hypothesis?
• RQ2 Can a canonical system of differential equations be constructed unconditionally for ω > 1 from the Riemann xi-function using integral operators?
• RQ3 How does the inner function property of Θω(z) encode the truth of the Riemann Hypothesis via the zeros of Aω(z)?
• RQ4 What is the spectral interpretation of the canonical system in terms of Hamiltonian matrices and positive semidefiniteness?
• RQ5 Can the construction of the canonical system be extended to all ω > 0 unconditionally, and what would that imply for RH?

Key findings

• For ω > 1, the canonical system associated with Θω(z) is constructed unconditionally using Fredholm determinants of compact integral operators.
• The function Θω(z) is a meromorphic inner function in the upper half-plane if and only if the Riemann Hypothesis holds.
• The L²-integrability of x−1/2 − hω⟨1⟩(x) on (1,∞) is equivalent to Θω(z) being an inner function in the upper half-plane.
• The construction of the canonical system via Mellin-Barnes integrals and residue calculus ensures that the associated Hamiltonian matrices are positive semidefinite if and only if RH holds.
• The paper provides a criterion for RH based on the positive semidefiniteness of the family of Hamiltonian matrices parametrized by ω > 0.
• The method extends Burnol’s approach on the gamma function to the Riemann xi-function, enabling unconditional construction for ω > 1.