Continued-fraction expansions for the Riemann zeta function and polylogarithms
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- by Djurdje Cvijović and Jacek Klinowski PDF
- Proc. Amer. Math. Soc. 125 (1997), 2543-2550
Abstract:
It appears that the only known representations for the Riemann zeta function $\zeta (z)$ in terms of continued fractions are those for $z=2$ and 3. Here we give a rapidly converging continued-fraction expansion of $\zeta (n)$ for any integer $n\geq 2$. This is a special case of a more general expansion which we have derived for the polylogarithms of order $n$, $n\geq 1$, by using the classical Stieltjes technique. Our result is a generalisation of the Lambert-Lagrange continued fraction, since for $n=1$ we arrive at their well-known expansion for $\log (1+z)$. Computation demonstrates rapid convergence. For example, the 11th approximants for all $\zeta (n)$, $n\geq 2$, give values with an error of less than 10$^{-9}$.References
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Additional Information
- Djurdje Cvijović
- Affiliation: Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom
- Email: dc133@cus.cam.ac.uk, d.cvijovic@usa.net
- Jacek Klinowski
- Affiliation: Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom
- Email: jk18@cam.ac.uk
- Received by editor(s): April 9, 1996
- Communicated by: Hal L. Smith
- © Copyright 1997 D. Cvijovic and J. Klinowski
- Journal: Proc. Amer. Math. Soc. 125 (1997), 2543-2550
- MSC (1991): Primary 11M99; Secondary 33E20
- DOI: https://doi.org/10.1090/S0002-9939-97-04102-6
- MathSciNet review: 1422859