New rapidly convergent series representations for $\zeta (2n+1)$
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- by Djurdje Cvijovic and Jacek Klinowski PDF
- Proc. Amer. Math. Soc. 125 (1997), 1263-1271 Request permission
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: \[ \zeta (2n+1)=(-1)^n\frac {4(2\pi )^{2n}}{(2n+1)!}\sum _{k=0}^\infty R_{2n+1,k}\zeta (2k),\qquad n\ge 1,\] 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.References
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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
- Received by editor(s): November 2, 1995
- Communicated by: Hal L. Smith
- © Copyright 1997 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 125 (1997), 1263-1271
- MSC (1991): Primary 11M99; Secondary 33E20
- DOI: https://doi.org/10.1090/S0002-9939-97-03795-7
- MathSciNet review: 1376755