  We consider a method for estimating the asymptotics of arithmetic functions
in the natural series in the paper. This question is closely related to the
distribution of primes in natural numbers. Problems often arise even with the
definition of the asymptotics of the mean value of arithmetic functions, and
even more so in the determination of the asymptotics of the moments of higher
orders. Therefore, the paper proposes a probabilistic approach, different from
the traditional one, based on the limiting distribution of arithmetic
functions. Arithmetic functions with limiting normal distribution are
considered. Several assertions are proved about the estimation of the
asymptotics of the moments of strongly additive arithmetic functions, as well
as additive functions of the class V and arithmetic functions of the class H
having such a limit distribution.
