bibkey: wunderlich2026bsyentropy authors: Henning Wunderlich year: 2026 title: Finite-radius Jensen and relative-entropy formulations of the Balazard–Saias–Yor criterion doi: null url: https://arxiv.org/abs/2610.10584v1 claim: The preprint gives finite-radius Jensen identities and a relative-entropy saturation form of the established Balazard–Saias–Yor criterion; it supplies no Robin divisor-sum estimate or FIB source bridge. strata_touched: [] license: citation-only triage: anchor

Finite-radius Jensen and relative-entropy formulations of the Balazard–Saias–Yor criterion

The source is arXiv:2610.10584v1, submitted 6 October 2026. The paper is a preprint; its cited Jensen–Hardy boundary identities were not independently audited here, and no Lean verification is claimed.

What is explicit

For $u(s)=(s-1)\zeta (s)$, the paper maps the unit disk to the left half-plane and applies Jensen’s formula at finite radius. In the normalization centered at $s=-1/2$, the mapped trivial zeros contribute a telescoping product. The residual is a nonnegative sum or integral over mapped nontrivial zeros off the critical line. The paper also gives a one-parameter gamma-quotient version.

The later construction starts from the established Balazard–Saias–Yor boundary family. Take $T$ with the specified density $6/[\pi (1+t^2)(4+t^2)]$ from (8.4), set $Z=|\Xi (T)/\Xi (0){|}^2$, and let $\mu$ be its law. With $M=\int z\mu (dz)$ from (8.5) and size-biased law $\nu (dz)=z\mu (dz)/M$, Theorem 8.2 states

$$D-D_{\mathrm{KL}}(\mu \Vert \nu )=4G\ge 0,$$

Here $D=\log (M\Xi (0{)}^2\Xi (2i{)}^2/\Xi (i{)}^4)$ is the constant from (8.12), and $G={\Delta }_1-\frac{1}{2}{\Delta }_2$ is the nonnegative off-critical zero correction from (7.9). RH is equivalent to saturation at $D$ for these particular measures; Remark 8.3 does not assert the bound for arbitrary probability pairs. The paper stresses that the data-processing equality is unconditional; the RH content is saturation of the external bound, not equality in data processing.

Boundary for the Robin/FIB route

This criterion controls a zero-side logarithmic integral. It does not identify that integral with

$$e^{\gamma }n\log \log n-\sigma (n)$$

at the same integer, nor does it provide a signed estimate for the project’s $\Phi$ or Möbius tail. A FIB address supplies an additive Zeckendorf source; this source and the address definitions supply no map to the law of $|\Xi (T){|}^2$ or the required boundary test function. Using the nonnegative Jensen defect as a Robin estimate therefore requires an additional proved correspondence.

The reusable research question is therefore an unproved interface: construct a common, source-preserving transform whose arithmetic projection is the Robin pointwise defect and whose analytic projection is this zero-side defect, with all endpoint and truncation errors retained. No such bridge is supplied by this preprint, so it is not a Robin or RH proof.