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

Author: 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 $n$, 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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Additional Information

H. M. Srivastava
Affiliation: Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia, Canada V8W 3P4

Keywords: Zeta functions, binomial theorem, Pochhammer symbol, functional equation, harmonic numbers, l'H\^{o}pital's rule, Bernoulli numbers, Euler polynomials, Euler's formula
Received by editor(s): June 2, 1997
Communicated by: Hal L. Smith
Article copyright: © Copyright 1999 American Mathematical Society

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