Some families of series representations for the Riemann \(\zeta(3)\) (Q1126951)

The authors give a very nice investigation of the series representations for \(\zeta(3)\), where \(\zeta(s)\) denotes the Riemann zeta function. Their systematic investigation rederives many formulae for \(\zeta(3)\) given by earlier authors as well as obtaining new formulae such as \[ \zeta(3)= {4\pi^2\over 13} \Biggl\{{3\over 4} -{1\over 2}\log\Biggl({2\pi\over 3}\Biggr)+ \sum^\infty_{k= 1} {\zeta(2k)\over k(2k+ 1)(2k+ 2)3^{2k}}\Biggr\}, \] which is analogous to the known result \(\zeta(3)= {\pi^2\over 6} \left\{{3\over 4}- {1\over 2}\log\left({\pi\over 3}\right)+ \sum^\infty_{k= 1}{\zeta(2k)\over k(2k+ 1)(2k+ 2)6^{2k}}\right\}\) due to \textit{N.-Y. Zhang} and \textit{K. S. Williams} [Rocky Mt. J. Math. 23, 1581-1592 (1993; Zbl 0808.11053)].