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 $$\gamma = \mathop {\li...
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| Date: | 1995 |
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| Main Authors: | , |
| Format: | Article |
| Language: | Russian English |
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Institute of Mathematics, NAS of Ukraine
1995
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/5435 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860511666495553536 |
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| author | Volchkov, V. V. Волчков, В. В. Волчков, В. В. |
| author_facet | Volchkov, V. V. Волчков, В. В. Волчков, В. В. |
| author_sort | Volchkov, V. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-19T08:59:05Z |
| description | We prove that the Riemann hypothesis on zeros of the zeta function ζ(s) is equivalent to the equality $$\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)$$ is the Euler constant. |
| first_indexed | 2026-03-24T03:16:31Z |
| format | Article |
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| id | umjimathkievua-article-5435 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T03:16:31Z |
| publishDate | 1995 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/df/48ffa6bd129d1f64dc87ae0dd78a38df.pdf |
| spelling | umjimathkievua-article-54352020-03-19T08:59:05Z On an equality equivalent to the Riemann hypothesis Об одном равенстве, эквивалентном гипотезе Римана Volchkov, V. V. Волчков, В. В. Волчков, В. В. We prove that the Riemann hypothesis on zeros of the zeta function ζ(s) is equivalent to the equality $$\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)$$ is the Euler constant. Доведено, іцо гіпотеза Рімана про нулі дзета-функції $ζ(s)$ зквівалентна рівності $$\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}},}$$ де $$\gamma = \mathop {\lim }\limits_{N \to \infty } \left( {\sum\limits_{n = 1}^N {\frac{1}{n} - \ln N} } \right)$$ — стала Пйлера. Institute of Mathematics, NAS of Ukraine 1995-03-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/5435 Ukrains’kyi Matematychnyi Zhurnal; Vol. 47 No. 3 (1995); 422–423 Український математичний журнал; Том 47 № 3 (1995); 422–423 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/5435/7559 https://umj.imath.kiev.ua/index.php/umj/article/view/5435/7560 Copyright (c) 1995 Volchkov V. V. |
| spellingShingle | Volchkov, V. V. Волчков, В. В. Волчков, В. В. On an equality equivalent to the Riemann hypothesis |
| title | On an equality equivalent to the Riemann hypothesis |
| title_alt | Об одном равенстве, эквивалентном гипотезе Римана |
| title_full | On an equality equivalent to the Riemann hypothesis |
| title_fullStr | On an equality equivalent to the Riemann hypothesis |
| title_full_unstemmed | On an equality equivalent to the Riemann hypothesis |
| title_short | On an equality equivalent to the Riemann hypothesis |
| title_sort | on an equality equivalent to the riemann hypothesis |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/5435 |
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