Wojciech Michalowski | Searcharxiv

arXiv2026

On the Pólya Frequency Order of the de Bruijn--Newman Kernel: Certified Failure at Order Five

We prove that the classical de Bruijn--Newman kernel $K(u)=Φ(|u|)$ is not a Pólya frequency function of order $5$ (PF$_5$). At $(u_0,h)=(0.01,0.05)$ we exhibit an explicit $5\times5$ Toeplitz minor whose determinant is rigorously enclosed in $[-1.8472496\times10^{-9},-1.8472225\times10^{-9}]$. The certificate uses 80-digit outward-rounded interval arithmetic and a proved truncation bound for the theta series. Eight further configurations are certified by both an explicit Leibniz expansion and an independent interval determinant computation. At the central configuration the determinants $D_2,D_3,D_4$ are positive, but this local sign pattern does not establish that the kernel is PF$_4$ globally. We also derive an exact finite formula for the first coefficient permitted by Vandermonde divisibility in the small-spacing expansion of $D_r(u_0,h)$. High-precision observations concerning the sign change of $C_5(u_0)$ and a Gaussian deformation are reported only as non-certified numerics. Version 2 withdraws the certified global sign and unique-threshold claims for $C_5$ made in version 1 because the derivative-tail enclosure was unsound; the direct PF$_5$ counterexample and its interval certificates are unaffected. The result concerns total positivity of this kernel and does not resolve the Riemann Hypothesis.

math.CA↗

arXiv2026

An explicit uniform cubic wedge for consecutive Toeplitz minors of the Riemann xi coefficients

Let (a_k) be the positive coefficient sequence of the normalized Riemann xi-function, and let D_{r,k} denote its consecutive Toeplitz minors. The Riemann Hypothesis is equivalent to (a_k) being a Polya frequency sequence of infinite order, and hence to nonnegativity of all Toeplitz minors. We prove that D_{r,k} > 0 for every r >= 2 and k >= 10^18 r^3. This gives an explicit cubic tail scale uniform in r, in contrast with Katkova's fixed-order asymptotic positivity. The proof does not use numerically verified zeros of the Riemann zeta-function. It combines a certified complex saddle-point analysis of the moment transform, an exact q-Pascal dilation semigroup controlling every degree simultaneously, and a weighted Banach-algebra majorant for the nonlinear remainder, closed by an inertia-preservation argument. All analytic constants are certified with directed rounding in Arb ball arithmetic, and all algebraic identities are verified in exact rational arithmetic. The ancillary files reproduce every certificate. The result concerns only the tail regime k much larger than r^3 and makes no progress on the Riemann Hypothesis, which concerns the complementary region.

math.NT↗