On values of the Riemann zeta function at positive integers (Q779834)

In the paper under review, the authors study the values of the Riemann zeta function at positive integers, giving new proofs of some known results related to them and obtaining some new results relating those values. To give a flavour of the paper, we quote some of their results. Write \( \zeta(2n)=\pi^{2n}\eta_n \). Then it is proved in Proposition~1 that \[ \sum_{n=0}^{\infty}\eta_nz^n=-\frac{1}{2}\sqrt{z}\cot\sqrt{z} \] and \[ \sum_{n=0}^{\infty}(-1)^n\eta_nz^n=-\frac{1}{2}\sqrt{z}\coth\sqrt{z}. \] An interesting consequence is the equation \[ \sum_{n=0}^{\infty}\frac{\zeta(2n)}{2^{2n}}=0. \] The Riemann-Lebesgue Lemma is applied to give recurrence relations for \( \zeta(2n) \) and \( \zeta(2n+1) \). A consequence of these relations is the following integral representation for the Apéry constant \( \zeta(3) \): \[ \zeta(3)=\frac{8}{7}\int_{0}^{\pi/2}x(2\log 2-x\cot x)\mathrm{d}x \] and a slightly more complicated one for \( \zeta(3) .\) Finally, a ``closed form'' expression for the series \( \sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k}\zeta(p+k) \) is given for any real \( p\geqslant 1. \)