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–New York–Oxford, 1988 . [12] Tamura , J. , Explicit formulae for Cantor series representing quadratic irrationals , Number Theory

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irrational values for rational values of the argument , Proc. Nat. Inst. Sci. India 13 ( 1947 ), 171 – 173 . [5] Duverney , D. , Nishioka , K

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Abstract  

We study the irrational factor function I(n) introduced by Atanassov and defined by

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$I(n) = \prod\nolimits_{\nu = 1}^k {p_\nu ^{1/\alpha _\nu } }$$ \end{document}
, where
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$n = \prod\nolimits_{\nu = 1}^k {p_\nu ^{\alpha _\nu } }$$ \end{document}
is the prime factorization of n. We show that the sequence {G(n)/n}n≧1, where G(n) = Πν=1 n I(ν)1/n, is convergent; this answers a question of Panaitopol. We also establish asymptotic formulas for averages of the function I(n).

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Summary We prove that, for any Tychonoff  X, the space C p(X) is K-analytic if and only if it has a compact cover {K p: p ? ??} such that K p subset K q whenever p,q ? ?? and p = q. Applying this result we show that if C p(X) is K-analytic then C p(?X) is K-analytic as well. We also establish that a space C p(X) is K-analytic and Baire if and only if X is countable and discrete.

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