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

Abstract

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 $Lz(a,b)$.