Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. B. C. Berndt, Ramanujan's Notebooks, Part II, Springer-Verlag, New York, 1989. MR 90b:01039
  • 2. O. Perron, Die Lehre von den Kettenbrüchen (3rd edition), Vol. I and II, Teubner, Stuttgart, 1954 and 1957. MR 16:239e; MR 19:25c
  • 3. H. S. Wall, Analytic Theory of Continued Fractions, Van Nostrand, New York, 1948. MR 10:32d
  • 4. W. B. Jones and W. J. Thron, Continued Fractions: Analytic Theory and Applications, Addison-Wesley, Reading, 1980. MR 82c:30001
  • 5. L. Lorentzen and H. Waadeland, Continued Fractions with Applications, North Holland, 1992. MR 93g:30007
  • 6. J. W. Bradshaw, Am. Math. Monthly, 51(1944) 389-391. MR 6:45c
  • 7. Yu. V. Nesterenko, Matem. Zametki, 59 (1996) 865-880.
  • 8. L. Lewin, Polylogarithms and Associated Functions, North-Holland, Amsterdam, 1981. MR 83b:33019
  • 9. A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev, Integrals and Series, Vols. 1 and 3, Gordon and Breach Science Publ., New York, 1986 and 1990. MR 88f:00013; MR 91c:33001
  • 10. P. Henrici, Applied and Computational Complex Analysis, Vol. 2, John Wiley, New York, 1977. MR 56:12235
  • 11. G. A. Baker, Jr. and P. Graves-Morris, Padé Approximants, Part I, Addison-Wesley, Reading, MA, 1981. MR 83a:41009a
  • 12. W. Rudin, Principles of Mathematical Analysis, McGraw Hill, New York, 1976. MR 52:5893
  • 13. A. Erédlyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Tables of Integral Transforms, Vol. II, McGraw-Hill, New York, Toronto and London, 1954. MR 16:468c
  • 14. G. Pólya and G. Szegö, Aufgaben und Lehrsätze aus der Analysis, Vol. 2 (reprinted), Dover Publications, New York, 1945. MR 7:418e
  • 15. V.V. Prasolov, Problems and Theorems in Linear Algebra, Am. Math. Society, Providence, Rhode Island, 1994. MR 95h:15002

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 11M99, 33E20

Retrieve articles in all journals with MSC (1991): 11M99, 33E20

Additional Information

Djurdje Cvijovic
Affiliation: Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom

Jacek Klinowski
Affiliation: Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom

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