Evaluation of the multiple zeta values $\zeta(2,\dots,2,3,2,\dots,2)$


A formula is given for the special multiple zeta values occurring in the title as rational linear combinations of products $\zeta(m)\pi^{2n}$ with $m$ odd. The existence of such a formula had been proved using motivic arguments by Francis Brown, but the explicit formula (more precisely, certain 2-adic properties of its coefficients) were needed for his proof in~\cite1 of the conjecture that all periods of mixed Tate motives over $\mathbb{Z}$ are $\mathbb{Q}[(2\pi i)^{\pm1}]$-linear combinations of multiple zeta values. The formula is proved indirectly, by computing the generating functions of both sides in closed form (one as the product of a sine function and a ${}_3F_2$-hypergeometric function, and one as a sum of 14 products of sine functions and digamma functions) and then showing that both are entire functions of exponential growth and that they agree at sufficiently many points to force their equality. We also show that the space spanned by the multiple zeta values in question coincides with the space of double zeta values of odd weight and find a relation between this space and the space of cusp forms on the full modular group.


Don Zagier

Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany