Evaluation of the multiple zeta values $\zeta(2,\dots,2,3,2,\dots,2)$
Evaluation of the multiple zeta values $\zeta(2,\dots,2,3,2,\dots,2)$
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DOI：
10.4007/annals.2012.175.2.11
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发表时间：
2012-03
影响因子：
4.9
通讯作者：
D. Zagier
中科院分区：
文献类型：
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作者：
D. Zagier
一个公式给出了特殊的多个zeta值出现在标题作为合理的线性组合的产品？(m)p 2n与m奇数。弗朗西斯·布朗已经用动机论证证明了这样一个公式的存在性，但是他在[1]中证明Z上的所有混合泰特动机的周期都是多个zeta值的Q[（2 pi）±1 ] -线性组合的猜想需要明确的公式（更准确地说，是其系数的某些2-adic性质）。该公式是间接证明，通过计算双方的生成函数在封闭形式（一个作为产品的正弦函数和3 F 2 -超几何函数，和一个作为总和的14个产品的正弦函数和digamma函数），然后表明，这两个是整个功能的指数增长，他们同意在足够多的点，迫使他们的平等。我们还表明，所涉及的多重zeta值所跨越的空间与奇数权的双zeta值空间重合，并找到了这个空间与满模群上的尖点形式空间之间的关系。
A formula is given for the special multiple zeta values occurring in the title as rational linear combinations of products ?(m)p 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 Z are Q[(2pi) ±1 ] -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 3 F 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.