Strict Log-Concavity of the Riemann Xi Kernel via a Curvature-Variance Decomposition and an Analytic Single-Crest Theorem for Planat's Envelope

Abstract

This preprint studies two analytic properties of the classical Riemann Xi kernel Φ and of the longitudinal envelope Qy introduced by Michel Planat in the context of paired theta-kernel methods. First, an independent and fully analytic proof of the strict log-concavity of the Riemann Xi kernel on the half-line is presented. The argument is based on an exact curvature-variance decomposition for the logarithmic second derivative of a positive theta-series. For the Riemann Xi kernel, this decomposition takes the form (logΦ)′′=−E[An]+Var(sn), where the first term measures the average logarithmic curvature of the theta components and the second measures the dispersion of their logarithmic slopes. Explicit elementary estimates yield the uniform quantitative bound (logΦ)′′(u)<−8,u≥0. Second, the longitudinal envelope Qy(a)=2∫0aΦ(a+b)Φ(a−b)cosh(2yb)db is shown, after symmetrization and the substitution x=2a, u=a+b, to admit the exact convolution representation Qy(x/2)=Fy∗F−y(x), where F±y(u)=1[0,∞)(u)Φ(u)e±yu. The quantitative log-concavity of Φ, together with preservation of strong log-concavity under convolution, then gives (logQy)′′(a)≤−16. Consequently, for every y>0, Qy has a unique global maximum across(y), with Qy′(a)>0for 0across(y). This provides an analytic proof of the single-crest property that is recorded numerically and identified as analytically open in Planat's June 2026 preprint. To the best of the author's knowledge, the specific convolution representation and its use to establish the single-crest theorem have not previously appeared in the literature. The manuscript also discusses its relation to recent work on the log-concavity of the Riemann Xi kernel, provides detailed proofs and quantitative estimates, and includes an appendix clarifying several notational points in the current version of Planat's preprint. This work does not constitute a proof of the Riemann Hypothesis.