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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

New rapidly convergent series representations for $\zeta (2n+1)$


Authors: Djurdje Cvijovic and Jacek Klinowski
Journal: Proc. Amer. Math. Soc. 125 (1997), 1263-1271
MSC (1991): Primary 11M99; Secondary 33E20
MathSciNet review: 1376755
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Abstract: We give three series representations for the values of the Riemann zeta function $\zeta (s)$ at positive odd integers. One representation extends Ewell's result for $\zeta (3)$ [Amer. Math. Monthly 97 (1990), 219-220] and is considerably simpler than the two generalisations proposed earlier. The second representation is even simpler:

\begin{displaymath}\zeta (2n+1)=(-1)^n\frac {4(2\pi )^{2n}}{(2n+1)!}\sum _{k=0}^\infty R_{2n+1,k}\zeta (2k),\qquad n\ge 1,\end{displaymath}

where the coefficients $R_{2n+1,k}$ for a fixed $n$ are rational in $k$ and are explicitly given by the finite sum involving the Bernoulli numbers. The third representation is obtained from the second by the Kummer transformation. We demonstrate the rapid convergence of this series using several examples.


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Additional Information

Djurdje Cvijovic
Affiliation: Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom
Email: dc133@cus.cam.ac.uk

Jacek Klinowski
Affiliation: Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom
Email: jk18@cus.cam.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9939-97-03795-7
PII: S 0002-9939(97)03795-7
Keywords: Riemann zeta function, series representations, rapidly convergent series.
Received by editor(s): November 2, 1995
Communicated by: Hal L. Smith
Article copyright: © Copyright 1997 American Mathematical Society