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 $u(s)=(s-1)\zeta (s)$ after mapping the unit disk onto $Res<1/2$. 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 $\Xi$-function. From the random variable $Z=|\Xi (T)/\Xi (0){|}^2$ we define its law $\mu$ and the size-biased law $\nu (\mathrm{d}z)=z\mu (\mathrm{d}z)/M$. The data-processing inequality is lossless for this statistic, and the Riemann hypothesis is equivalent to saturation of an explicit upper bound for $D_{\mathrm{KL}}(\mu \Vert \nu )$. 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.