On the Fractional Parts of Roots of Positive Real Numbers

DOI10.4169/AMER.MATH.MONTHLY.120.05.409zbMATH Open1305.11061arXiv1112.1759OpenAlexW2963302862WikidataQ58121935 ScholiaQ58121935MaRDI QIDQ2847007

Melvyn B. Nathanson

Publication date: 4 September 2013

Published in: American Mathematical Monthly (Search for Journal in Brave)

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

Full work available at URL: https://arxiv.org/abs/1112.1759

Mathematics Subject Classification ID

Arithmetic functions; related numbers; inversion formulas (11A25) Metric theory (11J83) Distribution modulo one (11J71) Arithmetic functions in probabilistic number theory (11K65)

Related Items (2)

The logic of fractional parts of real numbers ⋮ The sequence of fractional parts of roots

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