Arithmetic mean of differences of Dedekind sums

Emre Alkan, Maosheng Xiong, Alexandru Zaharescu

Research output: Contribution to journalArticlepeer-review


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 Qh(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 ℱQ, the set of Farey fractions of order Q in the unit interval [0,1] and a/q < a′/q′ are consecutive elements of ℱQ. We show that the limit lim Q→∞ A h (Q) exists and is independent of h.

Original languageEnglish (US)
Pages (from-to)175-187
Number of pages13
JournalMonatshefte fur Mathematik
Issue number3
StatePublished - Jul 2007


  • Dedekind sums
  • Farey fractions
  • Kloosterman sums

ASJC Scopus subject areas

  • General Mathematics


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