Arithmetic mean of differences of Dedekind sums

Abstract.

Recently, Girstmair and Schoissengeier studied the asymptotic behavior of the arithmetic mean of Dedekind sums $\frac{1}{\varphi(N)} \sum_{\mathop{\mathop{ 0 \le m< N}}\limits_{\gcd(m,N)=1}} \vert S(m,N)\vert$, as N → ∞. In this paper we consider the arithmetic mean of weighted differences of Dedekind sums in the form $A_{h}(Q)=\frac{1}{\sum_{\frac{a}{q} \in {\cal F}_{Q}}h\left(\frac{a}{q}\right)} \times \sum_{\frac{a}{q} \in {\cal F}_{\!Q}}h\left(\frac{a}{q}\right) \vert s(a^{\prime},q^{\prime})-s(a,q)\vert$, where $h:[0,1] \rightarrow {\Bbb C}$ is a continuous function with $\int_0^1 h(t) \, {\rm d} t \ne 0$, ${\frac{a}{q}}$ runs over ${\cal F}_{\!Q}$, the set of Farey fractions of order Q in the unit interval [0,1] and ${\frac{a}{q}}<\frac{a^{\prime}}{q^{\prime}}$ are consecutive elements of ${\cal F}_{\!Q}$. We show that the limit lim _Q→∞ A _h (Q) exists and is independent of h.