Applied Physics and Mathematics Annotation << Back ASYMPTOTICS OF THE NUMBER OF PRIMES AND SUMS OF FUNCTIONS OF PRIMES IN A SUBSET OF NATURAL NUMBERS V.L. VOLFSON Recently, with the development of computer technology and the Internet, the problem of the distribution of prime numbers has become of great practical importance, since it is directly related to the reliability of the so-called public key cryptographic systems. For example, the cryptographic strength of the currently widely used RSA encryption system is based on the computational complexity of factoring large natural numbers into prime factors. We define the asymptotic behavior "almost everywhere" of additive and multiplicative arithmetic functions in the paper. Classes of additive and multiplicative arithmetic functions are singled out for which the asymptotics coincides "almost everywhere" with the asymptotics of the corresponding strongly additive and multiplicative arithmetic functions. Several assertions are proved and examples are considered. Keywords: arithmetic function, additive arithmetic function, multiplicative arithmetic function, strongly additive arithmetic function, strongly multiplicative arithmetic function, probability space, analogue of the law of large numbers, asymptotics almost everywhere for arithmetic functions, normal order of an arithmetic function. DOI: 10.25791/pfim.06.2022.1245 Pp. 17-24.