A generating function of higher-dimensional Apostol–Zagier sums and its reciprocity law Original Research Article

We introduce higher-dimensional Dedekind sums with a complex parameter z, generalizing Zagierʹs higher-dimensional Dedekind sums. The sums tend to Zagierʹs higher-dimensional Dedekind sums as z→∞. We show that the sums turn out to be generating functions of higher-dimensional Apostol–Zagier sums which are defined to be hybrids of Apostolʹs sums and Zagierʹs sums. We prove reciprocity law for the sums. The new reciprocity law includes reciprocity formulas for both Apostol and Zagierʹs sums as its special case. Furthermore, as its application we obtain relations between special values of Hurwitz zeta function and Bernoulli numbers, as well as new trigonometric identities.