Finite-radius Jensen and relative-entropy formulations of the Balazard--Saias--Yor criterion
Abstract
We give two explicit reformulations of a Balazard--Saias--Yor type criterion for the Riemann hypothesis. First, Jensen's formula is applied to u(s)=(s−1)ζ(s) after mapping the unit disk onto Re s<1/2. For the normalization whose center corresponds to s=−1/2, the images zn=(4n−1)/(4n+3) of the trivial zeros yield a telescoping Jensen product. This isolates, at finite radius, the complete divergent contribution of the trivial zeros; the remaining defect is a nonnegative integrated count of mapped nontrivial zeros off the critical line. A one-parameter version gives a gamma-quotient product and an explicit finite part. Second, after passing through the established boundary identity, we combine two Poisson weights to obtain an absolutely convergent logarithmic integral of the Riemann Ξ-function. From the random variable Z=|Ξ(T)/Ξ(0)|2 we define its law μ and the size-biased law ν( d z)=zμ( d z)/M. The data-processing inequality is lossless for this statistic, and the Riemann hypothesis is equivalent to saturation of an explicit upper bound for DKL(μ|ν). We also record the associated Donsker--Varadhan and moment forms and express the target constant through Riemann's positive Fourier kernel. The analytic boundary criterion and Jensen--Hardy mechanism used here are known. The purpose of the paper is an explicit finite-radius treatment of the trivial zeros and an information-theoretic reformulation, not a new proof of the Riemann hypothesis or an independent boundary-integral criterion.
Related
- A Finite-Entropy Criterion for the Entropic Conditional Central Limit Theorem
- Finite attainment and efficient computation of regularized quantum channel relative entropy
- Isothermal walls in the compressible Navier-Stokes equations: affine entropy, ballistic energy, and entropy-stable discrete formulations
- Relative mean sensitivity and relative entropy under group actions
- Entropy-Regularized Mean-Field Games with Average-Reward Criterion: Span-Seminorm Contractions and Model-Free Learning
BibTeX
@misc{wunderlich2026finiteradius,
title = {{Finite-radius Jensen and relative-entropy formulations of the Balazard--Saias--Yor criterion}},
author = {Henning Wunderlich},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10584},
eprint = {2610.10584},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10584},
}
RIS
TY - UNPB
TI - Finite-radius Jensen and relative-entropy formulations of the Balazard--Saias--Yor criterion
AU - Henning Wunderlich
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.10584
UR - https://arxiv.org/abs/2610.10584
AB - We give two explicit reformulations of a Balazard--Saias--Yor type criterion for the Riemann hypothesis. First, Jensen's formula is applied to u(s)=(s−1)ζ(s) after mapping the unit disk onto Re s<1/2. For the normalization whose center corresponds to s=−1/2, the images z_n=(4n−1)/(4n+3) of the trivial zeros yield a telescoping Jensen product. This isolates, at finite radius, the complete divergent contribution of the trivial zeros; the remaining defect is a nonnegative integrated count of mapped nontrivial zeros off the critical line. A one-parameter version gives a gamma-quotient product and an explicit finite part. Second, after passing through the established boundary identity, we combine two Poisson weights to obtain an absolutely convergent logarithmic integral of the Riemann Ξ-function. From the random variable Z=|Ξ(T)/Ξ(0)|^2 we define its law μ and the size-biased law ν( d z)=zμ( d z)/M. The data-processing inequality is lossless for this statistic, and the Riemann hypothesis is equivalent to saturation of an explicit upper bound for D_KL(μ|ν). We also record the associated Donsker--Varadhan and moment forms and express the target constant through Riemann's positive Fourier kernel. The analytic boundary criterion and Jensen--Hardy mechanism used here are known. The purpose of the paper is an explicit finite-radius treatment of the trivial zeros and an information-theoretic reformulation, not a new proof of the Riemann hypothesis or an independent boundary-integral criterion.
ER -