Balazard--Saias--Yor准则的有限半径Jensen与相对熵形式
Finite-radius Jensen and relative-entropy formulations of the Balazard--Saias--Yor criterion
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中文总结 AI 辅助
本文给出黎曼假设的两种Balazard--Saias--Yor型准则的显式重述，通过有限半径处理平凡零点并结合信息论方法，未给出黎曼假设的新证明。
中文摘要 AI 辅助
本文给出了黎曼假设的两种显式重述形式，均为Balazard--Saias--Yor型准则。首先，在将单位圆盘映射到Re s<1/2后，将Jensen公式应用于u(s)=(s-1)ζ(s)；对于中心对应s=-1/2的归一化，平凡零点的像z_n=(4n-1)/(4n+3)产生一个望远镜Jensen乘积，这在有限半径处分离出平凡零点的完全发散贡献，剩余缺陷为临界线外映射非平凡零点的非负积分计数；单参数版本给出伽马商乘积和显式有限部分。其次，在通过已建立的边界恒等式后，结合两个泊松权重得到黎曼Ξ函数的绝对收敛对数积分；从随机变量Z=|Ξ(T)/Ξ(0)|²定义其分布μ和大小偏置分布ν(dz)=zμ(dz)/M；数据处理不等式对该统计量无损，且黎曼假设等价于D_KL(μ∥ν)显式上界的饱和；本文还记录了相关的Donsker--Varadhan形式和矩形式，并通过黎曼正傅里叶核表达目标常数。所用的解析边界准则和Jensen--Hardy机制为已知内容，本文目的是对平凡零点进行显式有限半径处理并给出信息论重述，并非黎曼假设的新证明或独立边界积分准则。
英文摘要
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 $ν(\,\mathrm d z)=zμ(\,\mathrm d z)/M$. The data-processing inequality is lossless for this statistic, and the Riemann hypothesis is equivalent to saturation of an explicit upper bound for $D_{\rm KL}(μ\|ν)$. 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.
