Experimental Mathematics

On the evaluation of Euler sums

Joe P. Buhler and Richard E. Crandall


Euler studied double sums of the form $$ \z(r,s)=\sum_{1\le m<n}{1\over n^sm^r} $$ for positive integers $r$ and $s$, and inferred, for the special cases $r = 1$ or $r+s$ odd, elegant identities involving values of the Riemann zeta function. Here we establish various series expansions of $\z(r, s)$ for real numbers $r$ and $s$. These expansions generally involve infinitely many zeta values. The series of one type terminate for integers $r$ and $s$ with $r+s$ odd, reducing in those cases to the Euler identities. Series of another type are rapidly convergent and therefore useful in numerical experiments.

Article information

Experiment. Math., Volume 3, Issue 4 (1994), 275-285.

First available in Project Euclid: 24 March 2003

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11M41: Other Dirichlet series and zeta functions {For local and global ground fields, see 11R42, 11R52, 11S40, 11S45; for algebro-geometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72}
Secondary: 11Y35: Analytic computations

Riemann zeta function periodic zeta function Lerch-Hurwitz zeta function polylogarithms incomplete gamma function


Crandall, Richard E.; Buhler, Joe P. On the evaluation of Euler sums. Experiment. Math. 3 (1994), no. 4, 275--285. https://projecteuclid.org/euclid.em/1048515810

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