Arithmetic of higher dimensional Dedekind–Rademacher sums

TextIn this paper we investigate higher order dimensional Dedekind–Rademacher sums given by the expression1a0m0+1∑k=1a0−1∏j=1dcot(mj)(πajka0), where a0a0 is a positive integer, a1,…,ada1,…,ad are positive integers prime to a0a0 and m0,m1,…,mdm0,m1,…,md 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 m0=⋯=md=0m0=⋯=md=0, we derive the reciprocity and rationality theorems of Zagier.VideoFor a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=J1_5H28fgAg.