The Ramanujan Journal

, Volume 16, Issue 1, pp 7–24 | Cite as

A new method in the study of Euler sums

  • Ankur Basu


A new method in the study of Euler sums is developed. A host of Euler sums, typically of the form \(\sum_{n=1}^{\infty}\frac{f(n)}{n^{s}}\sum_{m=1}^{n}\frac{g(m)}{m^{t}}\) , are expressed in closed form. Also obtained as a by-product, are some striking recursive identities involving several Dirichlet series including the well-known Riemann Zeta-function.


Riemann Zeta function Euler sums Recursions formulas 

Mathematics Subject Classification (2000)

40A25 40B05 11M99 33E99 


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  1. 1.
    Apostol, T.M., Vu, T.H.: Dirichlet Series related to the Riemann Zeta function. J. Number Theory 19, 85–120 (1984) MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bailey, D.H., Borwein, J.M., Girgensohn, R.: Experimental evaluation of Euler sums. Exp. Math. 3, 17–30 (1994) MATHMathSciNetGoogle Scholar
  3. 3.
    Basu, A., Apostol, Tom M.: A new method for investigating Euler sums. Ramanujan J. 4, 397–419 (2000) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Crandall, R.E., Buhler, J.P.: On the evaluation Euler sums. Exp. Math. 3, 275–285 (1994) MATHMathSciNetGoogle Scholar
  5. 5.
    Ramanujan, S.: Note Books, vol. 2 (1957) Google Scholar
  6. 6.
    Williams, G.T.: A method of evaluating ζ(2n). Am. Math. Mon. 60, 19–25 (1953) MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.C/o Ranjit BasuChandannagar, Dist. HooghlyIndia

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