Dedekind sums with roots of unity and their reciprocity law (Q1429096)

Dedekind sums with roots of unity are defined by \[ s_n(\gamma;(k,h))= \sum^{k-1}_{a=0} \sum^{h-1}_{b=0} {b\over h} \exp(2\pi i\gamma(ha+ kb))\overline B_n\Biggl({a\over k}+{b\over h}\Biggr), \] where \(\gamma\) is real, \(h\), \(k\) are positive integers, and \(\overline B_n(\cdot)\) is the periodic Bernoulli function of order \(n\). The author derives a reciprocity law for these sums. When \(\gamma\) is an integer the sums and reciprocity law reduce to those introduced by the reviewer [Duke Math. J. 17, 147--157 (1950; Zbl 0039.03801)].