math.NT — Number Theory
Coefficient Dynamics and Consecutive Toeplitz Minors for the de Bruijn–Newman Family
We give a normalization-consistent coefficient formulation of the de Bruijn–Newman heat flow attached to the Riemann ξ-function. For the classical kernel Φ and the family Ht(z) = Z ∞ 0 e tu2 Φ(u) cos(zu) du, Ft(w) = Ht(i √ w) = X n≥0 qn(t)w n , we prove the exact evolution laws ∂tFt = 2F ′ t + 4wF′′ t , q(m) n (t) = (2n + 2m)! (2n)! qn+m(t). The coefficients have positive integral representations and the transformed entire function has order at most 1/2. We relate the zeros of Ft to the Pólya–frequency property of (qn(t)), derive the rectangular Jacobi–Trudi identity for the consecutive Toeplitz minors, and state carefully the strict consecutive-minor criterion needed to pass from these minors to total positivity. In particular, for this coefficient family the real-zero condition is equivalent to strict positivity of every consecutive minor Dt(n, r), not merely to nonnegativity of that selected family. This distinction avoids an invalid use of a weak version of Fekete’s criterion. We also correct the multiplicity bookkeeping for the squared-variable map and record a Chebyshev convolution identity without attributing to it an unjustified duplication of zeros. The paper proves no new unconditional upper bound for the de Bruijn–Newman constant and no proof of the Riemann hypothesis; its purpose is to provide an auditable set of exact