An expression for Riemann Siegel function
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929399394205696 |
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| author | de Reyna, Juan Arias |
| author_facet | de Reyna, Juan Arias |
| contents | There are many analytic functions $U(t)$ satisfying $Z(t)=2\Re\bigl\{ e^{i\vartheta(t)}U(t)\bigr\}$. Here, we consider an entire function $\mathop{\mathcal L}(s)$ such that $U(t)=\mathop{\mathcal L}(\frac12+it)$ is one of the simplest among them. We obtain an expression for the Riemann-Siegel function $Z(t)$ in terms of the zeros of $\mathop{\mathcal L}(s)$. Implicitly, the function $\mathop{\mathcal L}(s)$ is considered by Riemann in his paper on Number Theory.
Riemann spoke of having used an expression for $Ξ(t)$ in his demonstration that most of the non-trivial zeros of the zeta function lie on the critical line. Therefore, any expression deserves a study. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_17365 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An expression for Riemann Siegel function de Reyna, Juan Arias Number Theory Primary 11M06, Secondary 30D99 There are many analytic functions $U(t)$ satisfying $Z(t)=2\Re\bigl\{ e^{i\vartheta(t)}U(t)\bigr\}$. Here, we consider an entire function $\mathop{\mathcal L}(s)$ such that $U(t)=\mathop{\mathcal L}(\frac12+it)$ is one of the simplest among them. We obtain an expression for the Riemann-Siegel function $Z(t)$ in terms of the zeros of $\mathop{\mathcal L}(s)$. Implicitly, the function $\mathop{\mathcal L}(s)$ is considered by Riemann in his paper on Number Theory. Riemann spoke of having used an expression for $Ξ(t)$ in his demonstration that most of the non-trivial zeros of the zeta function lie on the critical line. Therefore, any expression deserves a study. |
| title | An expression for Riemann Siegel function |
| topic | Number Theory Primary 11M06, Secondary 30D99 |
| url | https://arxiv.org/abs/2406.17365 |