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arXiv · 1112.1759

On the fractional parts of roots of positive real numbers

Melvyn B. Nathanson
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Abstract

Let [θ] denote the integer part and θ the fractional part of the real number θ. For θ> 1 and {θ^{1/n}} \neq 0, define M_θ(n) = [1/{θ^{1/n}}]. The arithmetic function M_θ(n) is eventually increasing, and \lim_{n\rightarrow \infty} M_θ(n)/n = 1/\log θ. Moreover, M_θ(n) is "linearly periodic" if and only if \log θis rational. Other results and problems concerning the function M_θ(n) are discussed.

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Melvyn B. Nathanson. 2013-04-12. On the fractional parts of roots of positive real numbers. https://arxiv.org/abs/1112.1759

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