When reading about multiple zeta values, I often find the claim that the case of length two $$ \zeta(s_1, s_2)=\sum_{n>m \geq 1} \frac{1}{n^{s_1}m^{s_2}}, \qquad s_1 \geq 2, \quad s_2 \geq 1 $$ was first considered by Euler. Does anybody know what precisely he proved about these numbers?

## 1 Answer

Euler proved in 1775 in Meditationes circa singulare serierum genus ("*Meditations about a singular type of series*") that
$$\sum_{i+j=n,\,i\geq 2,\;j\geq 1}\zeta(i,j)=\zeta(n),$$
as a special case of a more general sum rule.

Notice that Euler's definition of the multiple-zeta-value is slightly different than in the OP ($n\geq m$ instead of $n>m$)

The classical theory of zeta and $L$-functionsin the Milan Journal of Mathematics. $\endgroup$