A generalization of Rademacher's reciprocity law (Q2839289)

Let \(h/k\) be a rational number with \((h,k)=1\), \(k>1\), and let \(a\) be a complex number. Define NEWLINE\[NEWLINE c_{a}\left(\frac{h}{k}\right)=k^a\sum_{m=1}^{k-1}\cot\left(\frac{\pi hm}{k}\right)\zeta\left(-a,\frac{m}{k}\right), NEWLINE\]NEWLINE where \(\zeta(s,m/k)\) is the Hurwitz zeta-function. We have NEWLINE\[NEWLINE c_{-1}\left(\frac{h}{k}\right)=2\pi s\left(\frac{h}{k}\right), NEWLINE\]NEWLINE where \(s\left(\frac{h}{k}\right)\) is the Dedekind sum. This paper generalizes Rademacher's reciprocity formula for the Dedekind sum to \(c_{a}\left(\frac{h}{k}\right)\).