Advertisement

The Ramanujan Journal

, Volume 15, Issue 3, pp 377–405 | Cite as

The evaluation of character Euler double sums

  • J. M. Borwein
  • I. J. ZuckerEmail author
  • J. Boersma
Article

Abstract

Euler considered sums of the form
$$\sum_{m=1}^{\infty}\frac{1}{m^{s}}\sum_{n=1}^{m-1}\frac{1}{n^{t}}.$$
Here natural generalizations of these sums namely
$$[p,q]:=[p,q](s,t)=\sum_{m=1}^{\infty}\frac{\chi_{p}(m)}{m^{s}}\sum_{n=1}^{m-1}\frac{\chi_{q}(n)}{n^{t}},$$
are investigated, where χ p and χ q are characters, and s and t are positive integers. The cases when p and q are either 1,2a,2b or −4 are examined in detail, and closed-form expressions are found for t=1 and general s in terms of the Riemann zeta function and the Catalan zeta function—the Dirichlet series L −4(s)=1s −3s +5s −7s +⋅⋅⋅ . Some results for arbitrary p and q are obtained as well.

Keywords

Euler sums Dirichlet characters Riemann zeta function 

Mathematics Subject Classification (2000)

11M41 33E20 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Basu, A., Apostol, T.M.: A new method for investigating Euler sums. Ramanujan J. 4, 397–419 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Borwein, J., Bailey, D.: Mathematics by Experiment: Plausible Reasoning in the 21st Century. AK Peters, Natick (2003) Google Scholar
  3. 3.
    Borwein, J., Bailey, D., Girgensohn, R.: Experimentation in Mathematics: Computational Paths to Discovery. AK Peters, Natick (2004) zbMATHGoogle Scholar
  4. 4.
    Euler, L.: Meditationes circa singulare serierum genus. Novi Commun. Acad. Sci. Petropol. 20, 140–186 (1775) Google Scholar
  5. 5.
    Jordan, P.F.: Infinite sums of psi functions. Bull. Am. Math. Soc. 79, 681–683 (1973) zbMATHCrossRefGoogle Scholar
  6. 6.
    Lewin, L.: Polylogarithms and Associated Functions. North-Holland, New York (1981) zbMATHGoogle Scholar
  7. 7.
    Nielsen, N.: Die Gammafunktion. Chelsea, New York (1965) Google Scholar
  8. 8.
    Sitaramachandrarao, R.: A formula of S. Ramanujan. J. Number Theory 25, 1–19 (1987) zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Terhune, D.: Evaluations of double L-values. J. Number Theory 105, 275–301 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Zucker, I.J., Robertson, M.M.: Some properties of Dirichlet L-series. J. Phys. A: Math. Gen. 9, 1207–1214 (1976) CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Faculty of Computing ScienceDalhousie UniversityHalifaxCanada
  2. 2.Wheatstone Physics LaboratoryKing’s CollegeLondonUK
  3. 3.Dept. of Mathematics and Computing ScienceEindhoven University of TechnologyEindhovenThe Netherlands

Personalised recommendations