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. The paper solves the problems of determining the asymptotics of the number of primes and the sums of functions of primes in a subset of the natural series that satisfies the conditions that the asymptotic density of the number of primes in this subset is constant and not equal to zero. Assertions about asymptotics of the sums of functions of primes in a subset of the natural series satisfying the indicated conditions are proved. Necessary and sufficient conditions for the existence of these asymptotics are also proved. Keywords: asymptotic density of the number of primes in a subset of the natural series, asymptotic estimate for the number of primes in a subset of the natural series, Riemann conjecture, Abel summation formula, asymptotic estimate for the sums of functions of primes in a subset of the natural series, necessary and sufficient conditions for the existence of asymptotics. DOI: 10.25791/pfim.03.2022.1232 Pp. 31-39.