On the zeta function values \(\zeta (2k+1)\), \(k=1, 2, \dots\) (Q1912630)
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scientific article; zbMATH DE number 878068
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | On the zeta function values \(\zeta (2k+1)\), \(k=1, 2, \dots\) |
scientific article; zbMATH DE number 878068 |
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On the zeta function values \(\zeta (2k+1)\), \(k=1, 2, \dots\) (English)
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4 December 1996
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Series for the Riemann zeta function \(\zeta (s)\) at the odd integers \(s \geq 3\) are derived by integrating the logarithm of the infinite product representation of \(t \cos t\). The simplest result is \[ \zeta (3) = {\pi^2 \over 14} \left\{ {3 \over 2} + \log {4 \over \pi} - 4 \sum_{m = 1}^\infty {(2^{2m - 1} - 1) \zeta (2m) \over 2^{4m} m(2m + 1) (2m + 2)} \right\}. \]
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odd arguments
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Riemann zeta function
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