2018/04/20 by Victor Volfson, Volfson, Victor
Mathematics · Computer Science · #advanced mathematical theories #Mathematical Control Systems and Analysis #Probability and Statistical Research
paper · pdf · doi:10.48550/arxiv.1804.07539
The aim of the paper is to study the limit distributions and the asymptotic\nbehavior of summation arithmetic functions. A probabilistic approach based on\nthe use of the axioms of probability theory is used for these purposes.\nSufficient conditions are proved under which these functions have a limiting\nnormal distribution. Arithmetic functions having a limiting normal distribution\nare found in the paper. The author investigates summation functions of a\ngeneral form and finds sufficient conditions under which they have a limiting\nnormal distribution. Examples of arithmetic functions that satisfy these\nrequirements are considered. We prove an ergodic theorem for summation\narithmetic functions, for which the sequence of random variables has the\nstationarity property in the broad sense. The asymptotics of the growth of the\ndeviation of the values of the arithmetic function from its mean value are\ninvestigated. It is shown that an equivalent formulation of the Riemann\nhypothesis for the Mertens function is satisfied almost everywhere.\n