关于$ζ(n)$的某些含Nielsen广义多项式的整表示及相应的剖分问题
On Some Integral Representation Of $ζ(n)$ Involving Nielsen's Generalized Polylogarithms And The Related Partition Problem
摘要：
本文研究了一类$ζ(2n+1)$乘积的单变量整表示，其中$ζ(z)$是Riemann zeta函数，$n$是正整数.这样的表示涉及到积分$Lz(a,b):=\frac{1}{(a-1)!b!}\int_{0}^{1}\log^a (t)\log^b (1-t)dt/t$与正整数$a,b$，这是相关的尼尔森的广义多项式。通过分析相关的划分问题，讨论了这种整表示的结构，特别是用有限的$\mathbb{Q}(π)$-$Lz(a,b)$的线性组合表示$ζ(2n+1)$的积的条件.
英文摘要：
In this paper, we study a family of single variable integral representations for some products of $ζ(2n+1)$, where $ζ(z)$ is Riemann zeta function and $n$ is positive integer. Such representation involves the integral $Lz(a,b):=\frac{1}{(a-1)!b!}\int_{0}^{1}\log^a (t)\log^b (1-t)dt/t$ with positive integers $a,b$, which is related to Nielsen's generalized polylogarithms. By analyzing the related partition problem, we discuss the structure of such integral representation, especially the condition of expressing products of $ζ(2n+1)$ by finite $\mathbb{Q}(π)$-linear combination of $Lz(a,b)$.
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