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Arithmetic mean of differences of Dedekind sums. (English) Zbl 1203.11037
Summary: Recently, Girstmair and Schoissengeier studied the asymptotic behavior of the arithmetic mean of Dedekind sums $\frac{1}{\varphi(N)} \sum_{\mathop{\mathop{0 \leq m< N}}\limits_{\gcd(m,N)=1}} | S(m,N)|,$ as $$N \rightarrow \infty$$. In this paper we consider the arithmetic mean of weighted differences of Dedekind sums in the form $A_{h}(Q)=\frac{1}{\sum_{\frac{a}{q} \in {\mathcal F}_{Q}}h\left(\frac{a}{q}\right)} \times \sum_{\frac{a}{q} \in {\mathcal F}_{Q}}h\left(\frac{a}{q}\right) | s(a^{\prime},q^{\prime})-s(a,q)|,$ where $$h:[0,1] \rightarrow {\mathbb C}$$ is a continuous function with $$\int_0^1 h(t)\,dt \neq 0, {\frac{a}{q}}$$ runs over $${\mathcal F}_{Q}$$, 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 $${\mathcal F}_{Q}$$. We show that the limit $$\lim_{Q \rightarrow \infty} A_{h}(Q)$$ exists and is independent of $$h$$.

##### MSC:
 11F20 Dedekind eta function, Dedekind sums 11B57 Farey sequences; the sequences $$1^k, 2^k, \dots$$
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##### References:
 [8] Rademacher H, Grosswald E (1972) Dedekind Sums. Math Assoc Amer · Zbl 0251.10020
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