Natural divisors and the Brownian motion
On the space of integers \(1\leq m\leq x\), each having probability \(1/x\), we consider the random variables \[ X_x(m,t) =(\log \log x)^{-1/2} \biggl(\log_2 \tau\bigl(m, \exp (\log x)^t-t \log \log x\bigr) \biggr) \] as a random process, with \(t\) changing from 0 to 1. Here \(\tau (m,u)\) is the number of divisors of \(m\) up to \(u\). This process is shown to converge (as \(x\to\infty)\) to the Wiener process. This continues the author's investigations started in [Acta Math. Hung. 67, 1-17 (1995; Zbl 0823.11045)]. A version of the arcsine law is also proved. An arcsine law for the divisors was obtained by \textit{J.-M. Deshouillers}, \textit{F. Dress} and \textit{G. Tenenbaum} [Acta Arith. 34, 273-285 (1979; Zbl 0408.10035)], but this one is based on a different model and is essentially analogous to the arcsine law of \textit{P. Erdös} [Aequationes Math. 2, 177-183 (1969; Zbl 0174.08104)] for prime divisors.
scientific article; zbMATH DE number 912504
scientific article