bibkey: planatsole2026secondlevel
authors: Michel Planat and Patrick Solé
year: 2026
title: Second-Level Concavity of the Riemann Xi Kernel
doi: null
url: https://arxiv.org/abs/2608.19160v1
claim: Theorem 1.1 states strict log-concavity of the first Laguerre expression for the actual square-variable Riemann theta kernel on the positive half-line.
strata_touched:
  - D5/S3/Zeros/Jensen/SourceThetaMomentBounds
  - D5/S3/Analytic/Fourier/XiThetaTransform
license: citation-only
triage: anchor

Second-level concavity of the actual theta kernel

The pinned preprint uses Csordas’ normalization, denoted here by $K$ to distinguish it from the repository’s Romik kernel:

$$K(u)=\sum _{m\ge 1}\pi m^2(2\pi m^2{e}^{4u}-3)e^{5u-\pi m^2{e}^{4u}}.$$

Equation (1.2) sets $s(t)=K(\sqrt{t})$ and $f(t)=s^{\prime }(t{)}^2-s(t)s^{\prime \prime }(t)$. Theorem 1.1, printed p.1, states

$$f(t)>0,(\log f{)}^{\prime \prime }(t)<0(t>0),$$

where positivity of $f$ is the previously established first-level square-variable log-concavity. The second inequality addresses the Csordas–Dimitrov conjecture restated as Csordas Open Problem 4.14. This preprint-reported result should be reused rather than treated as a new theorem to discover in the project.

The proof uses a directed-rounding local Taylor certificate near $t=0$. Its two continuations use, respectively, a first theta summand with explicit tail bounds and a Jacobi differential/elliptic phase-space argument. They share the local certificate. Sections 3 and 8 describe the certificate bounds and supplementary scripts. These are the preprint’s computer-assisted proof claims; the scripts have not been independently executed here, and no Lean proof of Theorem 1.1 is supplied.

Correspondence with the fixed repository source

The source formulas in Romik’s kernel note give the normalization correspondence

$$\Phi (x)=2K(x/2),T=x^2/4,S(T)=\Phi (2\sqrt{T})=2s(T).$$

The paper’s $t$ is thus the fixed receiving coordinate $T$. Replacing $T$ by $x^2$ while retaining the same prescribed source polynomial nodes changes the correspondence and is not an admissible application of this result.

For $v>0$ and ${\xi }_0=\xi (1/2)>0$, the density-ratio notation is

$$r_v(T)=A_v{e}^{T/v}S(T),A_v=2\sqrt{\pi v}/{\xi }_0.$$

The elementary transport identity is

$$r_v^{\prime }(T{)}^2-r_v(T)r_v^{\prime \prime }(T)=A_v^2{e}^{2T/v}(S^{\prime }(T{)}^2-S(T)S^{\prime \prime }(T)).$$

The expression on the right is $4A_v^2{e}^{2T/v}f(T)$, so the source’s second-level conclusion transports as

$$\frac{d^2}{d{T}^2}\log (r_v^{\prime }(T{)}^2-r_v(T)r_v^{\prime \prime }(T))=(\log f{)}^{\prime \prime }(T)<0(T>0,v>0).$$

Multiplication by a positive constant and an exponential adds only an affine term to the logarithm. This is a paper-level application with the stated coordinate and tilt, not new mathematics or an additional formalized declaration.

Unpaid interface to the RH and Robin goals

Theorem 1.1 states a finite-order differential inequality for every $T>0$. A derivation of all higher derivative inequalities or a weighted comparison at every prescribed source Jensen node $v=n\lambda$ has not been established here. The neighboring-row signed estimate, and the extra vanishing needed at common zeros with correct multiplicities, remain separate obligations. The preprint explicitly makes no assertion of RH; this note supplies no proof of strict Robin for every integer greater than 5040.

The original-variable derivative counterexample concerns Csordas Open Problem 4.13, a different assertion. Its negative witness must not be transferred to $T$ derivatives without a new argument.

The separate proposed cumulative Peano sufficient condition is investigated in the private experiment repository’s theta-node interval certificate. That entry contains the source correspondence, analytic truncation bounds, executable certificate and its result. It tests a stronger condition than the original neighboring-row estimate and is not a theorem attributed to this preprint or a Lean proof of the target estimate.

The separate square-variable derivative-Hankel certificate investigates the stronger full alternating determinant hierarchy for the actual $S(T)$. It computes $T$ derivatives directly with analytic remainder bounds, rather than transferring an original-variable counterexample. The experiment’s conclusion concerns that sufficient hierarchy; it neither changes this preprint’s finite-order statement nor supplies or refutes the original prescribed-node scalar estimate.