发表机构
Dalian University of Technology（大连理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了Riemann xi-函数泰勒系数序列严格无限对数凹，解决了Zhu的猜想，方法结合复解析估计、区间算术与全局闭包论证。
AI 中文摘要
黎曼猜想等价于$F(x)$属于Laguerre--Pólya类。Brändén [J. Reine Angew. Math., 2011]证明了若Laguerre--Pólya类中的整函数具有非负泰勒系数，则其系数序列是无限对数凹的。因此，黎曼猜想蕴含$(\lambda_n)_{n\ge0}$的无限对数凹性。本文证明了序列$(\lambda_n)_{n\ge0}$是严格无限对数凹的。这解决了Zhu [Math. Z., 2023]的一个猜想。证明结合了迭代对数比率的显式复解析估计、有限指标范围的严格区间算术以及全局闭包论证。
英文摘要
The Riemann hypothesis is equivalent to $F(x)$ belonging to the Laguerre--Pólya class. Brändén [J. Reine Angew. Math., 2011] proved that if an entire function in the Laguerre--Pólya class has nonnegative Taylor coefficients, then its coefficient sequence is infinitely log-concave. Consequently, the Riemann hypothesis implies the infinite log-concavity of $(λ_n)_{n\ge0}$. In this paper, we prove that the sequence $(λ_n)_{n\ge0}$ is strictly infinitely log-concave. This resolves a conjecture of Zhu [Math. Z., 2023]. The proof combines explicit complex-analytic estimates for the iterated logarithmic ratios, rigorous interval arithmetic for a finite range of indices, and a global closure argument.
Comments60 pages
