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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2406.01474 |
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| _version_ | 1866913439004229632 |
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| author | de Reyna, J. Arias |
| author_facet | de Reyna, J. Arias |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_01474 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Report on some papers related to the function $\mathop{\mathcal R }(s)$ found by Siegel in Riemann's posthumous papers de Reyna, J. Arias Number Theory Primary 11M06, Secondary 30D99 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. |
| title | Report on some papers related to the function $\mathop{\mathcal R }(s)$ found by Siegel in Riemann's posthumous papers |
| topic | Number Theory Primary 11M06, Secondary 30D99 |
| url | https://arxiv.org/abs/2406.01474 |