A Counterexample to Csordas' Open Problem 4.13 for the Riemann Xi 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 isolated: the ninth-order inequality remains strictly false throughout an explicit neighborhood of the origin. The proof uses explicit derivative polynomials for the theta-series terms, rigorous rational interval estimates, and elementary bounds for exponential functions. All decisive sign determinations are reduced to exact integer or rational comparisons, without reliance on floating-point rounding. The result concerns a proposed family of kernel inequalities related to the classical Laguerre–Turán program. It does not imply that the Riemann Hypothesis is false and does not make any claim about the location of the zeros of the Riemann Xi function.