Strict Log-Concavity of the Laguerre Quantity for the Riemann Xi Kernel: Resolution of Csordas' Open Problem 4.14

This paper resolves Open Problem 4.14 posed by G. Csordas, concerning the Fourier kernel associated with the Riemann xi-function. The problem asks whether a certain log-concavity property holds for a quantity built from the derivatives of this kernel, a question tied to the classical Turan inequality program for the Riemann hypothesis. The proof splits the positive x-axis into two regions. On the unbounded region, a single leading term of the kernel's series expansion is shown to give an exact rational certificate with a large safety margin, and the effect of the remaining terms is bounded analytically. On the compact region, an eight-term symmetric core is expanded into a high-degree Taylor polynomial, converted exactly into Bernstein form, and every coefficient is verified to be positive using only integer and rational arithmetic. Analytic error bounds for the Taylor truncation and the omitted series tail are then shown to be far smaller than the certified margin. All sign decisions in the finite part of the argument reduce to integer or rational comparisons; no floating-point rounding is relied upon for correctness. Combined with an earlier known concavity result for the same kernel, the theorem also yields the double Turan inequalities through a known criterion of Csordas and Dimitrov. This is a necessary-condition result in the classical approach to the Riemann hypothesis. It does not prove the Riemann hypothesis, nor does it establish that all zeros of the Riemann xi-function are real. An accompanying reproducibility archive, referenced in the paper, contains scripts for independently rebuilding and verifying the exact rational and interval-arithmetic certificates described in the text.