黎曼的超论与函数论
Hyperg{é}om{é}trie et fonction z{ê}ta de Riemann
摘要：
我们证明了第二作者的“分母猜想”[40]关于zeta值中某些线性形式的系数的公因子。这些形式最近被构造以获得由$1,ζ(m),ζ(m+2),...,ζ(m+2h)$跨越的$\mathbb Q$上的向量空间的维数的下界，其中$m$和$h$是整数，使得$m\ge2$和$h\ge0$。特别地，我们立即得到以下结果作为推论：八个数$ζ(5),ζ(7),...,ζ(19)$中至少有一个是无理数，并且存在一个介于5和165之间的奇数$j$，使得1，$ζ(3)$和$ζ(j)$在$\mathbb{Q}$上线性无关。这分别加强了[41]和[8]中最近的一些结果。我们还证明了一个相关的猜想，由于瓦西里耶夫[49]，以及猜想，由于Zudilin [55]，对某些合理的近似$ζ(4)$。证明基于Andrews [3]的单和与多和之间的超几何恒等式。我们希望有可能将我们的构造应用于Zudilin [56]构造的更一般的线性形式，最终目的是加强他的结果，即其中一个数$ζ(5),ζ(7),ζ(9),ζ(11)$是无理数。
英文摘要：
We prove the second author's "denominator conjecture" [40] concerning the common denominators of coefficients of certain linear forms in zeta values. These forms were recently constructed to obtain lower bounds for the dimension of the vector space over $\mathbb Q$ spanned by $1,ζ(m),ζ(m+2),...,ζ(m+2h)$, where $m$ and $h$ are integers such that $m\ge2$ and $h\ge0$. In particular, we immediately get the following results as corollaries: at least one of the eight numbers $ζ(5),ζ(7),...,ζ(19)$ is irrational, and there exists an odd integer $j$ between 5 and 165 such that 1, $ζ(3)$ and $ζ(j)$ are linearly independent over $\mathbb{Q}$. This strengthens some recent results in [41] and [8], respectively. We also prove a related conjecture, due to Vasilyev [49], and as well a conjecture, due to Zudilin [55], on certain rational approximations of $ζ(4)$. The proofs are based on a hypergeometric identity between a single sum and a multiple sum due to Andrews [3]. We hope that it will be possible to apply our construction to the more general linear forms constructed by Zudilin [56], with the ultimate goal of strengthening his result that one of the numbers $ζ(5),ζ(7),ζ(9),ζ(11)$ is irrational.
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