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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | https://arxiv.org/abs/2406.18968 |
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| _version_ | 1866914850367602688 |
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| author | de Reyna, Juan Arias |
| author_facet | de Reyna, Juan Arias |
| contents | We apply Poisson formula for a strip to give a representation of $Z(t)$ by means of an integral. \[F(t)=\int_{-\infty}^\infty \frac{h(x)ζ(4+ix)}{7\coshπ\frac{x-t}{7}}\,dx, \qquad Z(t)=\frac{\Re F(t)}{(\frac14+t^2)^{\frac12}(\frac{25}{4}+t^2)^{\frac12}}.\] After that we get the estimate \[Z(t)=\Bigl(\frac{t}{2π}\Bigr)^{\frac74}\Re\bigl\{e^{i\vartheta(t)}H(t)\bigr\}+O(t^{-3/4}),\] with \[H(t)=\int_{-\infty}^\infty\Bigl(\frac{t}{2π}\Bigr)^{ix/2}\frac{ζ(4+it+ix)}{7\cosh(πx/7)}\,dx=\Bigl(\frac{t}{2π}\Bigr)^{-\frac74}\sum_{n=1}^\infty \frac{1}{n^{\frac12+it}}\frac{2}{1+(\frac{t}{2πn^2})^{-7/2}}.\] We explain how the study of this function can lead to information about the zeros of the zeta function on the critical line. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_18968 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Integral Representation for Riemann-Siegel $Z(t)$ function de Reyna, Juan Arias Number Theory Primary 11M06, Secondary 30D99 We apply Poisson formula for a strip to give a representation of $Z(t)$ by means of an integral. \[F(t)=\int_{-\infty}^\infty \frac{h(x)ζ(4+ix)}{7\coshπ\frac{x-t}{7}}\,dx, \qquad Z(t)=\frac{\Re F(t)}{(\frac14+t^2)^{\frac12}(\frac{25}{4}+t^2)^{\frac12}}.\] After that we get the estimate \[Z(t)=\Bigl(\frac{t}{2π}\Bigr)^{\frac74}\Re\bigl\{e^{i\vartheta(t)}H(t)\bigr\}+O(t^{-3/4}),\] with \[H(t)=\int_{-\infty}^\infty\Bigl(\frac{t}{2π}\Bigr)^{ix/2}\frac{ζ(4+it+ix)}{7\cosh(πx/7)}\,dx=\Bigl(\frac{t}{2π}\Bigr)^{-\frac74}\sum_{n=1}^\infty \frac{1}{n^{\frac12+it}}\frac{2}{1+(\frac{t}{2πn^2})^{-7/2}}.\] We explain how the study of this function can lead to information about the zeros of the zeta function on the critical line. |
| title | Integral Representation for Riemann-Siegel $Z(t)$ function |
| topic | Number Theory Primary 11M06, Secondary 30D99 |
| url | https://arxiv.org/abs/2406.18968 |