On the fractional parts of roots of positive real numbers

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.