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On the Pólya Frequency Order of the de Bruijn Newman Kernel. Certified Failure at Order Five and the Toeplitz Threshold Phenomenon

Wojciech Michałowski

Abstract

We prove that the classical de Bruijn--Newman kernel $K(u) = Φ(|u|)$, arising in the study of the Riemann zeta function via the de Bruijn--Newman constant, is not a Pólya frequency function of order $5$ (PF$_5$). The proof is computational: we exhibit an explicit $5 \times 5$ Toeplitz minor with rigorously certified negative determinant, established through interval arithmetic at 80-digit precision with formally bounded truncation and rounding errors. At the same Toeplitz configuration we certify positivity of all minors of orders $2$, $3$, and $4$; this shows that the \emph{Toeplitz PF threshold} within the two-parameter family $D_r(u_0,h)$ (Definition 2.1) lies exactly at order $5$ for this configuration, while the global question $K \in \mathrm{PF}_4$ remains an open problem (Section 8). We develop a systematic Toeplitz reduction that collapses the $2r$-dimensional configuration space of the PF$_r$ condition to a two-parameter family $D_r(u_0,h)$ of Toeplitz determinants. An asymptotic analysis in the grid spacing $h \to 0$ reveals leading coefficients $C_r(u_0)$ whose signs govern the PFthreshold. We prove the algebraic decomposition \[ C_r(u_0) \;=\; \sum_{\substack{k_0,\ldots,k_{r-1}\ge 0 \\ k_0+\cdots+k_{r-1}=r(r-1)}} \!\!\Bigl(\prod_{i=0}^{r-1} a_{k_i}(u_0)\Bigr)\, \det\bigl[(i-j)^{k_m}\bigr]_{i,m=0}^{r-1}, \] where $a_k(u_0) = K^{(k)}(u_0)/k!$, and verify that $C_r(u_0) > 0$ for $r \le 4$ at all tested points while $C_5(u_0) < 0$ for $u_0 \in (0, u_0^*)$ with a critical threshold $u_0^* = 0.031139\ldots$ (computed by bisection to 15 digits; see Section 5). This sign pattern, together with the positivity of $C_6$ and $C_7$, reveals a localized failure mechanism specific to order $5$. As a further probe of this phenomenon, we study the Gaussian deformation $K_t(u) = e^{tu^2}Φ(|u|)$ and compute, for each counterexample configuration $(u_0,h)$, the minimal $t$ at which the PF$_5$ violation is healed. This \emph{Toeplitz PF$_5$ Gaussian threshold} $λ_5^*(u_0,h)$ is configuration-dependent and should not be confused with the de Bruijn--Newman constant $Λ$, which concerns the reality of zeros of $H_t$ rather than total positivity of $K_t$.

On the Pólya Frequency Order of the de Bruijn Newman Kernel. Certified Failure at Order Five and the Toeplitz Threshold Phenomenon

Abstract

We prove that the classical de Bruijn--Newman kernel , arising in the study of the Riemann zeta function via the de Bruijn--Newman constant, is not a Pólya frequency function of order (PF). The proof is computational: we exhibit an explicit Toeplitz minor with rigorously certified negative determinant, established through interval arithmetic at 80-digit precision with formally bounded truncation and rounding errors. At the same Toeplitz configuration we certify positivity of all minors of orders , , and ; this shows that the \emph{Toeplitz PF threshold} within the two-parameter family (Definition 2.1) lies exactly at order for this configuration, while the global question remains an open problem (Section 8). We develop a systematic Toeplitz reduction that collapses the -dimensional configuration space of the PF condition to a two-parameter family of Toeplitz determinants. An asymptotic analysis in the grid spacing reveals leading coefficients whose signs govern the PFthreshold. We prove the algebraic decomposition \[ C_r(u_0) \;=\; \sum_{\substack{k_0,\ldots,k_{r-1}\ge 0 \\ k_0+\cdots+k_{r-1}=r(r-1)}} \!\!\Bigl(\prod_{i=0}^{r-1} a_{k_i}(u_0)\Bigr)\, \det\bigl[(i-j)^{k_m}\bigr]_{i,m=0}^{r-1}, \] where , and verify that for at all tested points while for with a critical threshold (computed by bisection to 15 digits; see Section 5). This sign pattern, together with the positivity of and , reveals a localized failure mechanism specific to order . As a further probe of this phenomenon, we study the Gaussian deformation and compute, for each counterexample configuration , the minimal at which the PF violation is healed. This \emph{Toeplitz PF Gaussian threshold} is configuration-dependent and should not be confused with the de Bruijn--Newman constant , which concerns the reality of zeros of rather than total positivity of .
Paper Structure (28 sections, 11 theorems, 26 equations, 5 tables)

This paper contains 28 sections, 11 theorems, 26 equations, 5 tables.

Table of Contents

  1. Introduction
  2. The de Bruijn--Newman framework
  3. Pólya frequency functions
  4. Main results
  5. Toeplitz Reduction
  6. Asymptotic Expansion and the Threshold Coefficient
  7. Taylor expansion of the Toeplitz entries
  8. Leading power and the threshold coefficient
  9. Computational Verification
  10. Rigorous interval arithmetic certification
  11. Additional certified counterexamples
  12. Systematic scan of $C_r(u_0)$
  13. Convergence of the $C_r$ extrapolation
  14. Algebraic verification of $C_5$
  15. The Failure Mechanism
  16. ...and 13 more sections

Key Result

Theorem 1.1

The de Bruijn--Newman kernel $K(u) = \Phi(|u|)$ is not PF$_5$. Specifically, the $5 \times 5$ Toeplitz matrix with $u_0 = 0.01$ and $h = 0.05$, satisfies This is certified by interval arithmetic: the rigorous enclosure is

Theorems & Definitions (21)

  • Theorem 1.1: PF$_5$ failure
  • Theorem 1.2: Positivity of lower-order Toeplitz minors
  • Remark 1.3
  • Theorem 1.4: Asymptotic threshold
  • Definition 2.1: Toeplitz determinant
  • Proposition 2.2: Dimension reduction
  • Lemma 3.1: Vandermonde divisibility
  • proof
  • Proposition 3.2
  • proof
  • ...and 11 more