RECIPROCITY LAWS FOR DEDEKIND COTANGENT SUMS

Abstract

Let $a_0,\dots ,a_d$ be pairwise coprime positive integers, $m_0,\dots ,m_d$ nonnegative integers, and $z_0,\dots ,z_d$ complex numbers. We study expressions of the form

$$\frac{1}{a_0^{m_0+1}}\sum _k\prod _{j=1}^d{\text{cot}}^{(m_j)}\pi \left(a_j\frac{k+z_0}{a_0}-z_j\right),$$

where the summation is taken over all $k(\text{mod }{a}_0)$ for which the summand is well defined. These sums generalize and unify various arithmetic sums introduced and studied by Dedekind, Apostol, Carlitz, Zagier, Berndt, Meyer, Sczech, Dieter and Beck. Special cases of these sums appear in various areas such as analytic and algebraic number theory, topology, algebraic and combinatorial geometry, and algorithmic complexity. In this paper, without any additional assumption on the parameters $a_0,\dots ,a_d$ and $z_0,\dots ,z_d$, we present a simple proof for the reciprocity formula for these generalized Dedekind cotangents sums. We recover the previous known results and improve them. As applications, we give explicit formulae for sums of secant and cosecant values in terms of Apostol–Bernoulli numbers.