A Counterexample to Csordas' Open Problem 4.13 for the Riemann ξ-Kernel

Abstract

This preprint gives a negative answer to Csordas’ Open Problem 4.13 concerning derivative Laguerre inequalities for the Jacobi theta kernel associated with the Riemann xi-function. At the symmetry point, the proposed inequalities are proved to hold through derivative order eight and to fail for the first time at order nine. The failure is not confined to the symmetry point: the same inequality is shown to remain strictly false throughout the explicit interval from minus 0.001 to 0.001. The proof is self-contained apart from standard structural facts about the xi-kernel. It uses explicit derivative polynomials, elementary exponential estimates, and exact rational interval arithmetic. The finite sign certificates are specified directly in the paper, and no floating-point computation or numerical rounding is used in any decisive sign determination. The result concerns a proposed family of inequalities for the xi-kernel and does not imply that the Riemann hypothesis is false.