On Some Integral Representation Of $\zeta(n)$ Involving Nielsen's Generalized Polylogarithms And The Related Partition Problem

In this paper, we study a family of single variable integral representations for some products of $\zeta(2n+1)$, where $\zeta(z)$ 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 $\zeta(2n+1)$ by finite $\mathbb{Q}(\pi)$-linear combination of $Lz(a,b)$.