Barnes' double zeta function, the Dedekind sum and Ramanujan's formula (Q1890456)

The author uses Barnes' double zeta function to obtain a new proof by contour integration of the reviewer's reciprocity law for generalized Dedekind sums [\textit{M. Apostol}, Duke Math. J. 17, 147--157 (1950; Zbl 0039.03801)] and, more generally, for what he calls Apostol-Rademacher Dedekind sums, given by \[ s_p(h,k; x,y)= \sum^{k-1}_{\nu= 0}\overline B_1\Biggl({y+\nu\over k}\Biggr)\overline B_p\Biggl({h\over k} (y+\nu)+ x\Biggr), \] where \(\overline B_p(\cdot)\) is a periodic Bernoulli function and \(p\geq 1\) is an odd integer. A limiting case gives a formula of Ramanujan expressing the Riemann zeta function \(\zeta(p)\) for odd \(p\geq 3\) in terms of Lambert series.