The First Failure of the Derivative Laguerre Inequalities for the Riemann Xi Kernel

Abstract

This preprint studies the derivative Laguerre inequalities for the standard kernel associated with the Riemann Xi function, in connection with Csordas’ Open Problem 4.13. The paper proves that the inequalities hold globally for the first eight derivative shifts. It then shows that the ninth shift is the first one to fail and determines its complete real sign pattern. The ninth-order quantity has exactly two real zeros, symmetric about the origin. It is strictly negative between these two zeros and strictly positive outside them. The positive transition point is isolated within a rigorous explicit interval. The proof combines derivative-polynomial representations of the theta-series terms, exact Bernstein-basis certificates on a compact region, and explicit analytic bounds for the remaining theta tail. All decisive sign determinations are reduced to exact integer or rational comparisons, and an accompanying verification archive independently reproduces the computational certificates. The results extend the author's earlier counterexample to Csordas’ Open Problem 4.13 by identifying the exact initial range of derivative shifts for which the proposed inequalities remain valid and by completely describing the first failing case. These kernel inequalities are related to the classical Laguerre–Turán program for the Riemann Xi function. The results do not prove or disprove the Riemann Hypothesis and make no claim about the location of the nontrivial zeros of the Riemann zeta function.