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The Riemann $Ξ$-function from primitive Markovian cycles I: A canonical construction

Douglas F. Watson

TL;DR

The paper constructs two canonical analytic objects from primitive Markov dynamics on discrete cycles: a PF$_\infty$ logarithmic kernel $\Phi$ whose bilateral Laplace transform has a Schoenberg--Edrei--Karlin factorization, and an Archimedean completion yielding a kernel whose Mellin transform matches the Riemann $Ξ$-function. By fixing a self-dual normalization via Jacobi inversion, it derives a Theta-kernel identity and defines an Archimedean completion operator $\mathcal{A}$ that recovers the classical $\Theta$-kernel and its Mellin counterpart $Ξ$. The total-positivity framework produces a canonical Laguerre--Pólya function $\Psi$ with real zeros and a reflection-symmetric Sell-like factorization $\mathcal{B}\Phi=E/\Psi$, while the Archimedean Mellin identification proves $F_{\mathrm{arch}}(z)=Ξ(2z)$, tying the construction to the classical zeta-analytic objects. An open “bridge” problem remains: whether the Laguerre--Pólya datum $\Psi$ can be identified with $Ξ(2\cdot)$ up to a zero-free factor, suggesting a deeper rigidity between PF$_\infty$ data and the $Ξ$-function. Together, these results provide an Archimedean, prime-free route to the $Ξ$-function and illuminate how locality, symmetry, and scaling can generate theta-series and real-zero structures from primitive dynamics.

Abstract

Starting from finite, local, reversible Markov dynamics on discrete cycles, we construct a scaling-limit renormalized trace kernel admitting an exact theta-series representation. The construction is entirely Archimedean and uses no Euler products, primes, or arithmetic spectral input. From this limit we define a logarithmic kernel $Φ$ and prove that it lies in the Pólya frequency class $\mathrm{PF}_\infty$, yielding via the Schoenberg-Edrei-Karlin classification a canonical Laguerre-Pólya function $Ψ$. Independently, we introduce an Archimedean completion operator and show that, at a self-dual normalization, the completed kernel coincides with the classical theta kernel, whose Mellin transform is the Riemann $Ξ$-function. We isolate a single remaining analytic problem relating $Ψ$ to $Ξ(2\cdot)$.

The Riemann $Ξ$-function from primitive Markovian cycles I: A canonical construction

TL;DR

The paper constructs two canonical analytic objects from primitive Markov dynamics on discrete cycles: a PF logarithmic kernel whose bilateral Laplace transform has a Schoenberg--Edrei--Karlin factorization, and an Archimedean completion yielding a kernel whose Mellin transform matches the Riemann -function. By fixing a self-dual normalization via Jacobi inversion, it derives a Theta-kernel identity and defines an Archimedean completion operator that recovers the classical -kernel and its Mellin counterpart . The total-positivity framework produces a canonical Laguerre--Pólya function with real zeros and a reflection-symmetric Sell-like factorization , while the Archimedean Mellin identification proves , tying the construction to the classical zeta-analytic objects. An open “bridge” problem remains: whether the Laguerre--Pólya datum can be identified with up to a zero-free factor, suggesting a deeper rigidity between PF data and the -function. Together, these results provide an Archimedean, prime-free route to the -function and illuminate how locality, symmetry, and scaling can generate theta-series and real-zero structures from primitive dynamics.

Abstract

Starting from finite, local, reversible Markov dynamics on discrete cycles, we construct a scaling-limit renormalized trace kernel admitting an exact theta-series representation. The construction is entirely Archimedean and uses no Euler products, primes, or arithmetic spectral input. From this limit we define a logarithmic kernel and prove that it lies in the Pólya frequency class , yielding via the Schoenberg-Edrei-Karlin classification a canonical Laguerre-Pólya function . Independently, we introduce an Archimedean completion operator and show that, at a self-dual normalization, the completed kernel coincides with the classical theta kernel, whose Mellin transform is the Riemann -function. We isolate a single remaining analytic problem relating to .
Paper Structure (34 sections, 30 theorems, 99 equations, 2 tables)

This paper contains 34 sections, 30 theorems, 99 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Primitive dynamical model
  3. Map of the proof
  4. Acknowledgements
  5. Self-dual Archimedean completion and forced normalization
  6. The self-dual scale is forced by Jacobi inversion
  7. The Archimedean completion operator
  8. Identification with the classical Theta at the self-dual scale
  9. Logarithmic kernel and exact reflection symmetry
  10. Definition of the logarithmic kernel and an exact mixture formula
  11. Quantitative tails and integrability
  12. Bilateral Laplace transform: convergence and analyticity
  13. Self-duality transport: twisted symmetry of Phi
  14. Exact Laplace reflection symmetry
  15. Total positivity and canonical Laguerre--Pólya factorization
  16. ...and 19 more sections

Key Result

Theorem 1.5

Assume the standing assumptions of Remark rem:standing-assumptions, so that the scaling-limit kernel $K_L$ exists. Then there exists a nonnegative kernel $\Phi\in L^1(\mathbb{R})$ such that:

Theorems & Definitions (84)

  • Remark 1.1: Translation-invariant specialization
  • Definition 1.2: Scaling--limit completed trace kernel
  • Definition 1.3: Archimedean (half--density) completion
  • Remark 1.4: Standing assumptions and existence of the scaling limit
  • Theorem 1.5: Structural output from primitive cycles
  • Theorem 1.6: Archimedean Mellin identification
  • Theorem 3.1: Theta-series form of the scaling-limit trace
  • Remark 3.2: Harmless reparameterization to Jacobi's standard form
  • Lemma 3.3: Forced self-dual normalization
  • proof
  • ...and 74 more