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Correlation of fractions with divisibility constraints
Author(s) -
Xiong Maosheng,
Zaharescu Alexandru
Publication year - 2011
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200710185
Subject(s) - combinatorics , mathematics , sequence (biology) , limiting , divisibility rule , function (biology) , square (algebra) , euler's totient function , euler's formula , mathematical analysis , geometry , chemistry , mechanical engineering , biochemistry , evolutionary biology , engineering , biology
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\documentclass{article}\usepackage{amsmath}\usepackage{amssymb}\usepackage{mathrsfs}\usepackage{bm} \begin{document}\pagestyle{empty}$$ {\mathscr{F}_{{B}\!,_Q}(I)}=\left\lbrace a/q \in I: 1 \le a \le q \le Q, \gcd (a,q)=\gcd (q,B_{Q}\!)=1 \right\rbrace . $$\end{document} 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