| For positive integers $s_1,\ldots,s_k$ with $s_1\ge 2$, the series $$ \sum_{n_1>\cdots>n_k\ge 1} n_1^{-s_1}\cdots n_k^{-s_k} $$ converges and its sum is denoted by $\zeta(s_1,\ldots,s_k)$. In case $k=1$ this number is nothing else than the value of the Riemann zeta function at the point $s_1$. From L.~Euler we know that for any integer $n\ge 1$, the value $\zeta(2n)$ is a rational multiple of $\pi^{2n}$. In 1978 R.~Apéry proved the irrationality of $\zeta(3)$. In May 2000 T.~Rivoal proved that infinitely many values $\zeta(2n+1)$, $n\ge 1$, are irrational. More precisely, for $n\ge N(\epsilon)$, among the $n+1$ numbers $1,\zeta(3),\zeta(5),\ldots,\zeta(2n+1)$, at least $(1-\epsilon)(\log n)/(1+\log 2)$ are linearly independent. This includes all known results dealing with the arithmetic nature of these numbers. In this survey we investigate the algebraic relations between the numbers $\zeta(s_1,\ldots,s_k)$ from a conjectural point of view. |
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