A conjecture of R. R. Hall on Farey points

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Abstract

Let I = (α, β] ⊆ (0, 1] be an interval, let γ1, < γ2 < ⋯ < γNI(Q) denote the Farey fractions of order Q from I and set Sr(Q) = ∑j=1NI(Q)j+1 - γj)(γj+r+1 - γj+r) for r ≧ 0. We prove that, for all r ≧ 2, there exists a constant Ir such that the following asymptotic formula holds: Sr(Q) = Sr(Q, (0,1)] = 6Ir2 Q2 + Or (log 1/(r+3)Q/Q2+1/(r+3) This implies, for all h ≧ 3, the existence of a constant D(h) such that ∑j=1N(0,1)(Q)j+hj)2 = 12(2h-1) logQ/π2 Q2 + D(h)/Q2 + Oh(log1/(h+2) Q/Q2+1/(h+2)) providing a positive answer to a conjecture of R. R. Hall. Furthermore, we prove that, for any r ≧ 2 and any such interval I, the sums Sr(Q, I) and Sr(Q) are related by Sr(Q, I) = |I|Sr(Q) + Oε(Q-2-1/(2r+4)+ε). For r ∈ {0, 1} we prove, for any I with rational end points, the existence of a constant cI such that S0(Q,I) = |I|S0(Q) + 2CI/Q2 + Oε(Q-2-1/10+ε) and S1(Q,I) = |I|SI(Q) + CI/Q2 + Oε(Q-2-1/12+ε).

Original languageEnglish (US)
Pages (from-to)207-236
Number of pages30
JournalJournal fur die Reine und Angewandte Mathematik
Volume535
DOIs
StatePublished - 2001
Externally publishedYes

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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