Investigations of the limit distribution and the asymptotic behavior of summation arithmetic functions

Abstract

The aim of the paper is to study the limit distributions and the asymptotic behavior of summation arithmetic functions. A probabilistic approach based on the use of the axioms of probability theory is used for these purposes. Sufficient conditions are proved under which these functions have a limiting normal distribution. Arithmetic functions having a limiting normal distribution are found in the paper. The author investigates summation functions of a general form and finds sufficient conditions under which they have a limiting normal distribution. Examples of arithmetic functions that satisfy these requirements are considered. We prove an ergodic theorem for summation arithmetic functions, for which the sequence of random variables has the stationarity property in the broad sense. The asymptotics of the growth of the deviation of the values of the arithmetic function from its mean value are investigated. It is shown that an equivalent formulation of the Riemann hypothesis for the Mertens function is satisfied almost everywhere.