Publications de l'Institut Mathématique, Nouvelle Série Vol. 97(111), pp. 139–147 (2015) 

IRRATIONALITY MEASURES FOR CONTINUED FRACTIONS WITH ARITHMETIC FUNCTIONSJaroslav Hancl, Kalle LeppäläDepartment of Mathematics and Centre of Excellence IT4Innovation, division of UO, Institute for Research and Applications of Fuzzy Modeling, University of Ostrava, Ostrava, Czech Republic; Department of Mathematical Sciences, University of Oulu, Oulu, FinlandAbstract: Let $f(n)$ or the base$2$ logarithm of $f(n)$ be either $d(n)$ (the divisor function), $\sigma(n)$ (the divisorsum function), $\varphi(n)$ (the Euler totient function), $\omega(n)$ (the number of distinct prime factors of $n$) or $\Omega(n)$ (the total number of prime factors of $n$). We present good lower bounds for $\bigl\frac MN\alpha\bigr$ in terms of $N$, where $\alpha=[0;f(1),f(2),\dots]$. Keywords: continued fraction, arithmetic functions, measure of irrationality Classification (MSC2000): 11J82; 11J70 Full text of the article: (for faster download, first choose a mirror)
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