An Elementary Derivation of Finite Cotangent Sums

Abstract

Finite cotangent sums are closely related to values of the Riemann zeta function at the positive even integers. Over the past decades, numerous authors have obtained explicit evaluations of these sums. In this note, we present an intuitive evaluation using Laurent coefficients of powers of the cotangent function. In fact, our expression is a concise restatement of a well-known formula by Berndt and Yeap that was obtained using contour integration.

MSC::

• Primary 11L03
• Secondary 11B68

Acknowledgment

The author is grateful to Jaebum Sohn, Xiaoyu He, and Vivian Kuperberg for their advice on structuring the presentation. The author is also thankful to the referees and the editor for their helpful comments that led to an improvement of this note.

Notes

1 Expand the numerator of $\text{sin}\hspace{0.17em}(nx)=(e^{2inx}-1)/(2i{e}^{\mathit{inx}})$ using the nth roots of unity.

2 Express the exponential definition of the cotangent function using the exponential generating function of the Bernoulli numbers. As $B_0=1,$ the summation can be taken from k = 0 to include the term $1/z.$