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| Formato: | Preprint |
| Publicado: |
2024
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2406.13278 |
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| _version_ | 1866909227797184512 |
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| author | de Reyna, Juan Arias |
| author_facet | de Reyna, Juan Arias |
| contents | Let $\mathop{\mathcal R}(s)$ be the function related to $ζ(s)$ found by Siegel in the papers of Riemann. In this paper we obtain the main terms of the mean values \[\frac{1}{T}\int_0^T |\mathop{\mathcal R}(σ+it)|^2\Bigl(\frac{t}{2π}\Bigr)^σ\,dt, \quad\text{and}\quad \frac{1}{T}\int_0^T |\mathop{\mathcal R}(σ+it)|^2\,dt.\] Giving complete proofs of some result of the paper of Siegel about the Riemann Nachlass. Siegel follows Riemann to obtain these mean values. We have followed a more standard path, and explain the difficulties we encountered in understanding Siegel's reasoning. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_13278 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Mean Values of the auxiliary function de Reyna, Juan Arias Number Theory Primary 11M06, Secondary 30D99 Let $\mathop{\mathcal R}(s)$ be the function related to $ζ(s)$ found by Siegel in the papers of Riemann. In this paper we obtain the main terms of the mean values \[\frac{1}{T}\int_0^T |\mathop{\mathcal R}(σ+it)|^2\Bigl(\frac{t}{2π}\Bigr)^σ\,dt, \quad\text{and}\quad \frac{1}{T}\int_0^T |\mathop{\mathcal R}(σ+it)|^2\,dt.\] Giving complete proofs of some result of the paper of Siegel about the Riemann Nachlass. Siegel follows Riemann to obtain these mean values. We have followed a more standard path, and explain the difficulties we encountered in understanding Siegel's reasoning. |
| title | Mean Values of the auxiliary function |
| topic | Number Theory Primary 11M06, Secondary 30D99 |
| url | https://arxiv.org/abs/2406.13278 |