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

   

 

Continued-fraction expansions
for the Riemann zeta function
and polylogarithms


Authors: Djurdje Cvijovic and Jacek Klinowski
Journal: Proc. Amer. Math. Soc. 125 (1997), 2543-2550
MSC (1991): Primary 11M99; Secondary 33E20
MathSciNet review: 1422859
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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$^{-9}$.


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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, 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

DOI: http://dx.doi.org/10.1090/S0002-9939-97-04102-6
PII: S 0002-9939(97)04102-6
Keywords: Riemann zeta function; polylogarithms; continued fractions.
Received by editor(s): April 9, 1996
Communicated by: Hal L. Smith
Article copyright: © Copyright 1997 D. Cvijovic and J. Klinowski