Interlacing and Derivative Log-Concavity for the Riemann–Jacobi Kernel

Abstract

This paper gives rigorous new proofs for the unresolved Coffey–Csordas Conjecture 2.5 concerning the derivative Turán inequalities of the Riemann–Jacobi kernel. Specifically, it proves the previously unresolved cases [n=2,3,4,5,] in addition to the known case (n=1). A central feature of the proof is that the finite compact-range verification is not based on floating-point numerics. It is carried out by the attached exact certificate coffey_csordas_cases_1_5_full_certificate.json, containing rigorous outward-rounded lower bounds on every certified subinterval for all five cases. The certificate is independently checkable by the accompanying exact-arithmetic verifier verify_coffey_csordas_cases_1_5.py. The verification uses rational arithmetic, fixed-point interval arithmetic with outward rounding, polynomial evaluation, rigorous alternating-series bounds for exponentials, and explicit theta-tail estimates. The paper also proves an unconditional all-order theorem for each individual theta term. After reduction to an explicit polynomial recursion, Rolle's theorem, an exact degree count, strict zero interlacing, and a Wronskian identity establish the derivative Turán inequalities at every order for a single theta term. Finally, the paper proves that the full Coffey–Csordas hierarchy is a sufficient differential condition for Hirschman–Widder bell-shape. Consequently, the rigorously verified cases (n=1,\ldots,5) imply that the full Riemann–Jacobi kernel has exactly (k) simple real zeros in its (k)-th derivative for (k=0,\ldots,5), with strict interlacing of consecutive derivatives. Supporting files include the full exact certificate coffey_csordas_cases_1_5_full_certificate.json and its independently executable verifier verify_coffey_csordas_cases_1_5.py.