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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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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$^{-9}$.
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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