We define a combinatorial game in R from which we derive numerous new inequalities between higher-dimensional Dedekind sums. Our approach is motivated by a recent article by Dilcher and Girstmair, who gave a nice probabilistic interpretation for the classical Dedekind sum. Here we introduce a game analogous to Dilcher and Girstmair’s model in higher dimensions.

We study the asymptotic behavior of the classical Dedekind sums s(sk/tk) for the sequence of convergents sk/tk k ≥ 0, of the transcendental number

In this paper we study the Euler polynomials and functions and derive some interesting formulae related to the Euler polynomials and functions. From those formulae we consider Dedekind type DC(Daehee-Changhee)sums and prove reciprocity laws related to DC sums.

Using the Euler-MacLaurin summation formula, we give alternative proofs for the reciprocity formulas of Apostol’s Dedekind sums and generalized Hardy-Berndt sums s3,p(b, c) and s4,p(b, c). We also obtain an integral representation for each sum.

Elliptic Dedekind sums were introduced by R. Sczech as generalizations of classical to complex lattices. We show that for any lattice with real $j$-invariant, the values suitably normalized elliptic are dense in num