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

TL;DR

This paper investigates the limit distributions and asymptotic behavior of summation arithmetic functions using probabilistic methods, establishing conditions for normal distribution limits and exploring implications related to the Riemann hypothesis.

Contribution

It provides new sufficient conditions for summation arithmetic functions to have a limiting normal distribution and proves an ergodic theorem in this context.

Findings

Identifies conditions under which summation arithmetic functions have a normal distribution limit

Proves an ergodic theorem for stationary summation functions

Shows an almost everywhere equivalence of the Riemann hypothesis for the Mertens function

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.