Moment integrals of \(1/\sin t\) and related zeta-values
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 \(\mathbb Z\), \(1 \leq j \leq r\).
scientific article; zbMATH DE number 6348220
scientific article