Arithmetic of higher dimensional Dedekind–Rademacher sums

Distribution and non-vanishing of special values of L-series attached to Erdős functions

“…It can be shown that the number given by ( 5) is in fact rational. For a proof of this fact and further properties of the higher dimensional Dedekind sums, see [27] and [2]. Moreover, these sums satisfy a reciprocity law given by:…”

“…In the course of evaluating moments of L(k, f ), we also encounter a generalization of the higher dimensional Dedekind sums which were introduced by D. Zagier [27]. These sums, studied by A. Bayad and A. Raouj [2], are defined as follows. For i = 0, ⋯, d, let a 0 be a positive integer, a 1 , ⋯, a d be positive integers co-prime to a 0 and m 0 , ⋯, m d be non-negative integers.…”

On generalized Dedekind sums involving quasi-periodic Euler functions

“…Dedekind [13] introduced the sum (1.1) in connection with the modular properties of the Dedekind η-function and deduced from his reciprocity law 12hk s(h, k) + s(k, h) = h 2 − 3hk + k 2 + 1 (1.3) (see [2, p. 62] and [19, p. 148]). In later years several mathematicians generalized s(h, k) and showed that the generalized functions too satisfy a reciprocity law, see [1][2][3][4][5][7][8][9][10][11][12]14,16,18,[20][21][22][23] and the references given there. The first proof of (1.3) does not employ the theory of the Dedekind η-function is due to Rademacher [17].…”

Multiple Dedekind–Rademacher sums in function fields

“…Bayad and Raouj [2] investigated multiple Dedekind-Rademacher sums, and they showed that these sums are rational and the reciprocity law holds. We next review the Petersson-Knopp identity.…”