Correlation of fractions with divisibility constraints

✍ Scribed by Maosheng Xiong; Alexandru Zaharescu

• Publisher: John Wiley and Sons
• Year: 2011
• Tongue: English
• Weight: 202 KB
• Volume: 284
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.200710185

✦ Synopsis

Abstract

Let \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 Q~1~ < Q~2~. For any subinterval I ⊂ [0, 1], let

It is shown that if B~Q~ ≪ Q^log 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