Approximate formula for $Z(t)$
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866909231753461760 |
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| author | de Reyna, Juan Arias |
| author_facet | de Reyna, Juan Arias |
| contents | The series for the zeta function does not converge on the critical line but the function \[G(t)=\sum_{n=1}^\infty \frac{1}{n^{\frac12+it}}\frac{t}{2πn^2+t}\] satisfies $Z(t)=2\Re\{e^{i\vartheta(t)}G(t)\}+O(t^{-\frac56+\varepsilon})$. So one expects that the zeros of zeta on the critical line are very near the zeros of $\Re\{e^{i\vartheta(t)}G(t)\}$. There is a related function $U(t)$ that satisfies the equality $Z(t)=2\Re\{e^{i\vartheta(t)}U(t)\}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_18150 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Approximate formula for $Z(t)$ de Reyna, Juan Arias Number Theory Primary 11M06, Secondary 30D99 The series for the zeta function does not converge on the critical line but the function \[G(t)=\sum_{n=1}^\infty \frac{1}{n^{\frac12+it}}\frac{t}{2πn^2+t}\] satisfies $Z(t)=2\Re\{e^{i\vartheta(t)}G(t)\}+O(t^{-\frac56+\varepsilon})$. So one expects that the zeros of zeta on the critical line are very near the zeros of $\Re\{e^{i\vartheta(t)}G(t)\}$. There is a related function $U(t)$ that satisfies the equality $Z(t)=2\Re\{e^{i\vartheta(t)}U(t)\}$. |
| title | Approximate formula for $Z(t)$ |
| topic | Number Theory Primary 11M06, Secondary 30D99 |
| url | https://arxiv.org/abs/2406.18150 |