On the fractional parts of roots of positive real numbers

TL;DR

This paper investigates the properties of the function M_{ heta}(n), related to the fractional parts of roots of positive real numbers, revealing its asymptotic behavior and conditions for linear periodicity.

Contribution

It introduces and analyzes the function M_{ heta}(n), establishing its growth rate and the criterion for linear periodicity based on the rationality of extlog heta.

Findings

M_{ heta}(n) is eventually increasing.

lim_{n o\infty} M_{ heta}(n)/n = 1/ extlog heta.

M_{ heta}(n) is linearly periodic iff extlog heta is rational.

Abstract

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