High-Order Center Failures of the Derivative Laguerre Inequalities for the Riemann Xi Kernel

Abstract

This preprint studies the high-order behavior of the derivative Laguerre inequalities for the standard Jacobi-theta kernel associated with the Riemann Xi function. The paper focuses on the symmetry point at the origin and derives a three-term asymptotic expansion for the even derivatives of the kernel after a natural normalization. The leading term is identified with an explicit Jacobi theta profile whose real zeros and sign pattern can be determined exactly. As a consequence, the paper proves that failures of the derivative Laguerre inequalities at the center occur infinitely often. It also determines the asymptotic number of such failures up to a given derivative order and gives an explicit asymptotic description of their locations. For a set of transition indices of natural density one, the corresponding failure order is determined by an exact floor formula involving an explicit transition constant. The proof combines the modular transformation of the Jacobi theta function, exact coefficient transfer through generalized Laguerre polynomials, a uniform Hermite–Volterra expansion in a growing logarithmic window, Jacobi product identities, and Weyl equidistribution. A supplementary reproducibility package contains the LaTeX source, verification scripts, numerical diagnostics, and machine-readable outputs. The numerical computations are included only as independent consistency checks and are not used in the proofs.