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
    $\begingroup$ This may fit better at hsm.stackexchange.com $\endgroup$ Feb 2, 2016 at 16:02
  • 14
    $\begingroup$ He was trying to understand $\zeta(3)$ after he discovered the rationality of $\zeta(2k)/\pi^{2k}$. He was unsuccessful with $\zeta(3)$ but was led to many identities among MZV (which arise naturally), the simplest being $\zeta(3) = \zeta(2,1)$. See the introduction (pages 11-13) of Bombieri's article The classical theory of zeta and $L$-functions in the Milan Journal of Mathematics. $\endgroup$ Feb 2, 2016 at 16:20

1 Answer 1


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 $n>m$)


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.