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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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