Finite cotangent sums and the Riemann zeta function (Q2777498)

The authors give a closed formula for the sums NEWLINE\[NEWLINE \sum _{p=0}^{q-1} \operatorname {cot}^n \left [\frac {(\xi +p)\pi} q\right], \qquad \sum _{p=1}^{q-1} \operatorname {cot}^{2n} \left (\frac {p\pi} q\right), NEWLINE\]NEWLINE where \(n\) and \(q\) are positive integers \((q\geq 2)\) and \(\xi \) is a non-integer real number. This leads to simplifications of \textit{T. Apostol}'s formula concerning the Riemann \(\zeta \)-functions; [see Am. Math. Mon. 80, 425-431 (1973; Zbl 0267.10050)].