Arithmetic of higher dimensional Dedekind–Rademacher sums

Abstract

Text In this paper we investigate higher order dimensional Dedekind–Rademacher sums given by the expression 1 a 0 m 0 + 1 ∑ k = 1 a 0 − 1 ∏ j = 1 d cot ( m j ) ( π a j k a 0 ) , where a 0 is a positive integer, a 1 , … , a d are positive integers prime to a 0 and m 0 , m 1 , … , m d are non-negative integers. We study arithmetical properties of these sums. For instance, we prove that these sums are rational numbers and we explicit good bounds for their denominators. A reciprocity law is given generalizing a theorem of Rademacher for the classical Dedekind sums and a theorem of Zagier for higher dimensional Dedekind–Rademacher sums. On the other hand, our reciprocity results can be viewed as complements to the Beck reciprocity theorem. Taking m 0 = ⋯ = m d = 0 , we derive the reciprocity and rationality theorems of Zagier. Video For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=J1_5H28fgAg.