## Experimental Mathematics

### On the evaluation of Euler sums

#### Abstract

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

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

Dates
First available: 24 March 2003

http://projecteuclid.org/euclid.em/1048515810

Mathematical Reviews number (MathSciNet)
MR1341720

Zentralblatt MATH identifier
0833.11045

#### Citation

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