Arithmetic mean of differences of Dedekind sums

Recently, Girstmair and Schoissengeier studied the asymptotic behavior of the arithmetic mean of Dedekind sums 1 / ℓ (N) ∑ 0 ≤ m < N _gcd(m,N)=1 |S(m,N)|, as N → ∞. In this paper we consider the arithmetic mean of weighted differences of Dedekind sums in the form A _h(Q)=1 / ∑_{a/q ∈ ℱ Q}h(a/q)× ∑_{a/q ∈Q} h(a/q) | s(a′,q′)-s(a,q)|, where h:[0,1] → ℂ is a continuous function with ∫₀¹ h(t)dt ≠ 0, a/q runs over ℱ_Q, the set of Farey fractions of order Q in the unit interval [0,1] and a/q < a′/q′ are consecutive elements of ℱ_Q. We show that the limit lim _Q→∞ A _h (Q) exists and is independent of h.