An analogue of the Dedekind–Rademacher sum and certain ray class invariants of real quadratic fields

Abstract

In this paper, we consider a certain product of double sine functions as an analogue of the Dedekind–Rademacher sum. Its reciprocity formulas are established by decomposition of a certain double zeta function. As their applications, we reconstruct and refine a part of Arakawaʼs work on ray class invariants of real quadratic fields, and prove directly explicit relations between various invariants which are defined in terms of the double sine function and are related to the Stark–Shintani conjecture. Moreover, in some examples, new expressions of the invariants are revealed. As two appendices, we give a new proof of Carlitzʼs three-term relation for the Dedekind–Rademacher sum and a simple proof of Arakawaʼs transformation formula for an analogue of the generalized Eisenstein series originated with Lewittes.