关于Siegel在黎曼遗稿中发现的函数$\mathop{\mathcal R }(s)$相关论文的报告
Report on some papers related to the function $\mathop{\mathcal R }(s)$ found by Siegel in Riemann's posthumous papers
AI总结：
本报告总结了多篇论文，旨在回答Siegel关于黎曼遗稿中函数$\mathop{\mathcal R }(s)$零点位置的问题，并改进其结果，但尚未证明黎曼的断言。
摘要：
在给Weierstrass的一封信中，Riemann断言，临界线上高度为$T$的$ζ(s)$零点数$N_0(T)$近似等于该高度下零点总数$N(T)$。Siegel研究了Riemann的一些遗稿，试图找到这一断言的证明。他发现了一个函数$\mathop{\mathcal R }(s)$，其零点与函数$ζ(s)$的零点相关。Siegel得出结论：Riemann的论文中不包含任何关于其断言证明的想法，并将$\mathop{\mathcal R }(s)$零点的位置与$ζ(s)$零点的位置联系起来，同时询问了$\mathop{\mathcal R }(s)$零点的位置。本文是对我们即将上传到arXiv的几篇论文的总结，在这些论文中，我们试图回答Siegel关于$\mathop{\mathcal R }(s)$零点位置的问题。这些文章还包含对Siegel结果的改进，以及证明Riemann断言的其他可能途径，但尚未实现这一目标。
英文摘要：
In a letter to Weierstrass Riemann asserted that the number $N_0(T)$ of zeros of $ζ(s)$ on the critical line to height $T$ is approximately equal to the total number of zeros to this height $N(T)$. Siegel studied some posthumous papers of Riemann trying to find a proof of this. He found a function $\mathop{\mathcal R }(s)$ whose zeros are related to the zeros of the function $ζ(s)$. Siegel concluded that Riemann's papers contained no ideas for a proof of his assertion, connected the position of the zeros of $\mathop{\mathcal R }(s)$ with the position of the zeros of $ζ(s)$ and asked about the position of the zeros of $\mathop{\mathcal R }(s)$. This paper is a summary of several papers that we will soon upload to arXiv, in which we try to answer Siegel's question about the position of the zeros of $\mathop{\mathcal R }(s)$. The articles contain also improvements on Siegel's results and also other possible ways to prove Riemann's assertion, but without achieving this goal.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
