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

TL;DR

This paper explores integral representations of odd zeta values involving Nielsen's polylogarithms and examines the partition problem related to expressing these values as rational linear combinations of specific integrals.

Contribution

It introduces a new integral representation for products of odd zeta values using Nielsen's generalized polylogarithms and analyzes the conditions for expressing these products as rational combinations of such integrals.

Findings

Derived integral representations for products of ζ(2n+1).

Analyzed the structure and conditions for expressing zeta products via finite rational combinations.

Connected the integral representations to the partition problem and algebraic structures.

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 $Lz(a,b)$.