Arithmetic mean of differences of Dedekind sums期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

Arithmetic mean of differences of Dedekind sums

Authors:	Emre Alkan Maosheng Xiong Alexandru Zaharescu

Affiliation:	(1) University of Illinois at Urbana-Champaign, Urbana, IL, USA

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 $A_{h}(Q)=frac{1}{sum_{frac{a}{q} in {cal F}_{Q}}hleft(frac{a}{q}right)} times sum_{frac{a}{q} in {cal F}_{!Q}}hleft(frac{a}{q}right) vert s(a^{prime},q^{prime})-s(a,q)vert$ , where 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.

Keywords:	2000 Mathematics Subject Classification: 11F20
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