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

2021/01/09 by Wang, Xiaowei
#11P81 #11Y60 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2101.03421

Abstract

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):=(1)/((a-1)!b!)∫₀¹log^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 ℚ(π)-linear combination of Lz(a,b).

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