Approximation by special values of harmonic zeta function and log-sine integrals (Q2511248)

The central objects in this paper are the two the log-sine integrals \[ I(m,a)=\int _{0}^{a}t^{m} \log \left(2\sin \frac{t}{2} \right)\, {\kern 1pt} d{\kern 1pt} t\, \text{ and}\, J(m,a)=\int _{0}^{a}t^{m} \log ^{2} \left(2\sin \frac{t}{2} \right)\, {\kern 1pt} d{\kern 1pt} t. \] The authors proved several interesting representations involving these integrals and the values of the Riemann zeta function \(\zeta (s)=\sum _{n=1}^{\infty }\frac{1}{n^{s} } \) , the harmonic (Euler) zeta function \(\zeta _{H} (s)=\sum _{n=1}^{\infty }\frac{H_{n} }{n^{s} } \) , and the odd-harmonic zeta function \(\zeta _{O} (s)=\sum _{n=1}^{\infty }\frac{O_{n} }{n^{s} } \) , where \(H_{n} =1+\frac{1}{2} + \cdots+\frac{1}{n} \) are the harmonic numbers and \(O_{n} =1+\frac{1}{3} +\cdots +\frac{1}{2n-1} \) are the odd harmonic numbers. The results are presented in five theorems. For example, Theorem 3 shows the representation by a finite sum and explicit coefficients \[ I(m+1,2\pi )-\pi I(m,2\pi )=\sum _{k=1}c(m,k)\left[\zeta _{H} (2k)-\zeta (2k+1)\right]. \] In order to obtain this representation the author starts from the Fourier expansion \[ \frac{1}{2} (t-\pi )\log \left(2\sin \frac{t}{2} \right)=\sum _{n=1}^{\infty }\left(\frac{H_{n} }{n} -\frac{1}{n^{2} } \right) \sin nt,\; 0<t<2\pi . \] Multiplying both sides by \(t^{m} \) and integrating termwise on \([0,2\pi ]\), he obtains an important recurrence formula which leads to the theorem. Similar ideas are used for the other results.