The p-adic analytic Dedekind sums

In this paper, using Cohen's and Tangedal and Young's theory on the p -adic Hurwitz zeta functions, we construct the analytic Dedekind sums on the p -adic complex plane C p . We show that these Dedekind sums interpolate Carlitz's higher order Dedekind sums p -adically. From Apostol's reciprocity law for the generalized Dedekind sums, we also prove a reciprocity relation for the special values of these p -adic Dedekind sums. Finally, the parallel results for the analytic Dedekind sums on the p -adic complex plane associated with Euler polynomials have also been given.