Multiplikative Funktionen auf kurzen Intervallen.
A model of the Brownian motion defined in terms of the natural divisors is proposed and weak convergence of the related measures in the space [0,1] is proved. An analogon of the Erdös arcsine law, known for the prime divisors [6] (see [14] for the proof), is obtained. These results together with the author’s investigation [15] extend the systematic study [9] of the distribution of natural divisors. Our approach is based upon the functional limit theorems of probability theory.
Let pₘ(n) stand for the middle prime factor of the integer n ≥ 2. We first establish that the size of log pₘ(n) is close to √(log n) for almost all n. We then show how one can use the successive values of pₘ(n) to generate a normal number in any given base D ≥ 2. Finally, we study the behavior of exponential sums involving the middle prime factor function.
A homothetic arithmetic function of ratio is a function such that for every . Periodic arithmetic funtions are always homothetic, while the converse is not true in general. In this paper we study homothetic and periodic arithmetic functions. In particular we give an upper bound for the number of elements of in terms of the period and the ratio of .