On an equality equivalent to the Riemann hypothesis
We prove that the Riemann hypothesis on zeros of the zeta function ζ(s) is equivalent to the equality $$\int\limits_0^\infty {\frac{{1 - 12t^2 }}{{(1 + 4t^2 )^3 }}dt} \int\limits_{1/2}^\infty {\ln |\varsigma (\sigma + it)|d\sigma = \pi \frac{{3 - \gamma }}{{32}},}$$ where $$\gamma = \mathop {\li...
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| Datum: | 1995 |
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| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | Russisch Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
1995
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/5435 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
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Ukrains’kyi Matematychnyi Zhurnal| Zusammenfassung: | We prove that the Riemann hypothesis on zeros of the zeta function ζ(s) is equivalent to the equality $$\int\limits_0^\infty {\frac{{1 - 12t^2 }}{{(1 + 4t^2 )^3 }}dt} \int\limits_{1/2}^\infty {\ln |\varsigma (\sigma + it)|d\sigma = \pi \frac{{3 - \gamma }}{{32}},}$$ where $$\gamma = \mathop {\lim }\limits_{N \to \infty } \left( {\sum\limits_{n = 1}^N {\frac{1}{n} - \ln N} } \right)$$ is the Euler constant. |
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