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ISSN 1088-6826(e) ISSN 0002-9939(p)

Some rapidly converging series for $\zeta (2n+1)$

Author(s): H. M. Srivastava
Journal: Proc. Amer. Math. Soc. 127 (1999), 385-396.
MSC (1991): Primary 11M06, 11M35, 33B15; Secondary 11B68, 33E20, 40A30
MathSciNet review: 1610797
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Abstract: For a natural number , the author derives several families of series representations for the Riemann Zeta function $\zeta (2n+1)$ . Each of these series representing $\zeta (2n+1)$ converges remarkably rapidly with its general term having the order estimate:

$\begin{equation*}O(k^{-2n-1}\cdot m^{-2k})\qquad (k\to \infty ;\quad m=2,3,4,6).\end{equation*}$

Relevant connections of the results presented here with many other known series representations for $\zeta (2n+1)$ are also pointed out.

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