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Natural divisors and the brownian motion

Eugenijus Manstavičius (1996)

Journal de théorie des nombres de Bordeaux

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 𝐃 [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.

Normal numbers and the middle prime factor of an integer

Jean-Marie De Koninck, Imre Kátai (2014)

Colloquium Mathematicae

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.

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