Moment integrals of \(1/\sin t\) and related zeta-values
type: Item
instance of: scholarly article
title: Moment integrals of \(1/\sin t\) and related zeta-values
zbMATH Open document ID: 1314.11053
author: Jun Furuya
author: Makoto Minamide
author: Yoshio Tanigawa
published in: The Ramanujan Journal
publication date: 2014-09-26T00:00:00Z
review text: Let \(I_n(x)\) be the moment integral of \(\frac{1}{\sin t}\), \(0<x<\pi\), \(n \in\mathbb Z_+\), defined by \[ I_n(x)=\int_{0}^{x}\frac{t^n}{\sin t} dt. \] In particular, \(I_1(\frac{\pi}{2})=2G\), where \(G\) is the Catalan constant. The function \(I_n(x)\) is a generalization of one of the integral representations of the constant \(G\). Let \(\zeta(s)\) and \(L(s,\chi)\) be the well-known Riemann zeta and Dirichlet \(L\)-functions defined for \(\sigma>1\), respectively; here \(s=\sigma+it\) is a complex variable, and \(\chi\) is a certain Dirichlet character. For \(n \in\mathbb Z_+\) and \(x \in\mathbb R\), \(0<x<\pi\), define \[ C_n(x)=\sum_{k=0}^{\infty}\frac{\cos(2k+1)x}{(2k+1)^n} \quad {\text{and}}\quad S_n(x)=\sum_{k=0}^{\infty}\frac{\sin(2k+1)x}{(2k+1)^n}. \] In the paper, as application of the moment integral of \(\frac{1}{\sin t}\), some results on the relations between the function \(I_n(x)\) and the functions \(\zeta(s)\), \(L(s,\chi_4)\), the multiple zeta-values \(\zeta_r(k_1,k_2,\ldots,k_r)\) and multiple \(L\)-values \(L_r(k_1,k_2,\ldots,k_r;f_1,f_2,\ldots,f_r)\) are discussed (here \(\chi_4\) is fixed non-principal Dirichlet character modulo 4). For example, for \(m \in\mathbb Z_+\) and \(0<x<\pi\), \[ I_{2m-1}\big(\frac{\pi}{2}\big)=2\sum_{l=0}^{m-1}\binom{2m-1}{2l+1}(2l+1)! (-1)^{l}\big(\frac{\pi}{2}\big)^{2m-2l-2}L(2l+2,\chi_4), \] and, in particular, for \(n \in\mathbb Z_+\) and \(0<x \leq \frac{\pi}{2}\), \[ I_n(\frac{\pi}{2})=2^n n!L_n(\underbrace{1,1,\ldots,1}_{n-1},2; \chi_4), \] when the multiple \(L\)-values \(L_r(k_1,k_2,\ldots,k_r;f_1,f_2,\ldots,f_r)\) is defined by \[ L_r(k_1,\ldots,k_r;f_1,\ldots,f_r)=\sum_{m_1=1}^{\infty}\ldots\sum_{m_r=1}^{\infty}\frac{f_1(m_1)\ldots f_r(m_r)}{m_1^{k_1}\ldots(m_1+\cdots+m_r)^{k_r}}, \] with period \(M\) and periodic functions \(f_j(m)\) on \(\mathbb Z\), \(1 \leq j \leq r\).
reviewed by: Roma Kačinskaitė
Mathematics Subject Classification ID: 11M06
Mathematics Subject Classification ID: 11M32
Mathematics Subject Classification ID: 11M41
zbMATH DE Number: 6348220
zbMATH Keywords: moment integral
zbMATH Keywords: Catalan constant
zbMATH Keywords: Dirichlet \(L\)-values
zbMATH Keywords: multiple \(L\)-values
zbMATH Keywords: Bernoulli numbers
MaRDI profile type: Publication
full work available at URL: s11139-013-9467-1
OpenAlex ID: W2111922390
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DOI: 10.1007/S11139-013-9467-1
description: scientific article; zbMATH DE number 6348220