发表机构
Université de Lorraine（洛林大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究提出一种更简洁的新证明，证实黎曼ζ函数超67.25%的非平凡零点是单零点且在临界线上，超83.62%互不相同，其核心是用希尔伯特空间不等式替代原有有限维矩阵框架。
AI 中文摘要
我们获得了一个全新的、概念上更简单的无条件证明，证明黎曼ζ函数的非平凡零点中超过67.25%是单零点且位于临界线上，且至少83.62%的非平凡零点是互不相同的。这些结果的证明最近由Anthropic开发的Claude内部研究版本完成，随后由Alpöge和Furman验证。该证明技术复杂，核心机制并不直观，它结合了线性代数的若干要素，包括Weil埃尔米特型的有限维矩阵表示、埃尔米特矩阵的秩-迹不等式，以及利用显式公式对零点进行的二阶矩计算。我们的新证明更简短，通过用希尔伯特空间不等式替代整个有限维矩阵框架来实现，这使得可以直接应用Montgomery关于ζ函数零点对关联的定理，该定理采用了Baluyot、Goldston、Suriajaya和Turnage-Butterbaugh得到的无条件形式。
英文摘要
We obtain a new, conceptually simpler, unconditional proof that more than $67.25\%$ of the non-trivial zeros of the Riemann zeta function are simple and on the critical line, and that at least $83.62\%$ of the non-trivial zeros are distinct. Our approach also yields two new unconditional estimates on simple zeros and zeros on the critical line. More precisely, we prove that the proportion of zeros that are simple or lie on the critical line (or both) is at least $88.76\%$, and that the average of the proportions of simple zeros and of zeros on the critical line is at least $83.62\%$. A proof of the bounds for simple zeros on the critical line and for distinct zeros was very recently produced by an internal research version of Claude developed by Anthropic and subsequently verified by two mathematicians at Anthropic, Levent Alpöge and Ralph Furman, whereas our two additional estimates are neither stated nor proved in the Claude paper. The argument produced by Claude is technically intricate, and its main mechanism is not immediately transparent. It combines several ingredients from linear algebra, including a finite-dimensional matrix representation of Weil's Hermitian form and a rank--trace inequality for Hermitian matrices, with a second moment calculation over the zeros using the explicit formula. Our new approach proceeds by replacing the entire finite-dimensional matrix framework by a single Hilbert space inequality, which allows for a direct application of Montgomery's theorem on the pair correlation of zeros of the zeta function, in the unconditional form obtained by Baluyot, Goldston, Suriajaya and Turnage-Butterbaugh.
Comments17 pages. We establish two new unconditional results: more than 88.76% of zeta zeros are simple or lie on the critical line (or both), and the average of the proportions of simple zeros and of zeros on the critical line is at least 83.62%. We also expand the exposition
