Correlation of fractions with divisibility constraints

Abstract

AbstractLet \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$ B=(B_{Q}\!)_{{Q \in {\mathbb N}}} $\end{document} be an increasing sequence of positive square free integers satisfying the condition that $ B_{{Q_1}}\vert B_{{Q_2}} $ whenever Q1 < Q2. For any subinterval I ⊂ [0, 1], let It is shown that if BQ ≪ Qlog log Q/4, then the limiting pair correlation function of the sequence \documentclass{article}\usepackage{amsmath,amssymb,mathrsfs,bm}\pagestyle{empty}\begin{document}$ ({\mathscr{F}_{{B}\!,_Q}(I)})_{Q \in {\mathbb N}} $\end{document} exists and is independent of the subinterval I. Moreover, the sequence is Poissonian if $ \lim_{Q \rightarrow \infty }{{\varphi (B_{Q}\!)}\over{B_{Q}\!}} = 0 $, and exhibits a very strong repulsion if $ \lim_{Q \rightarrow \infty }{{\varphi (B_{Q}\!)}\over{B_{Q}\!}} \ne 0 $, where φ is Euler's totient function. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim