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

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IRRATIONALITY MEASURES FOR CONTINUED FRACTIONS WITH ARITHMETIC FUNCTIONS

Jaroslav Hancl, Kalle Leppälä

Department 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, Finland

Abstract: Let $f(n)$ or the base-$2$ logarithm of $f(n)$ be either $d(n)$ (the divisor function), $\sigma(n)$ (the divisor-sum 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

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Electronic fulltext finalized on: 16 Apr 2015. This page was last modified: 21 Apr 2015.

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