Applied Physics and Mathematics Annotation << Back COMPARISON OF PROBABILISTIC AND EXACT METHODS FOR ESTIMATING THE ASYMPTOTIC BEHAVIOR OF SUMMATION ARITHMETIC FUNCTIONS V.L. VOLFSON The problem of distribution of prime numbers has acquired an important practical significance, since it is directly related to the reliability of so-called cryptographic systems with a public key in recent times, with the development of computer technology and the Internet. For example, the cryptographic resistance of the widely used RSA encryption system is based on the computational complexity of prime factorization of large natural numbers. In the paper we investigate the summation arithmetic functions and in particular the arithmetic function - the number of primes not exceeding a real number x. The paper compares probabilistic and exact methods for estimating the asymptotic behavior of summation arithmetic functions, and estimates of the results are obtained by precise methods. Conditions for stationarity in the broad sense are investigated for summation arithmetic functions. A lemma and theorems about the estimation of the standart deviation for the summation arithmetic Mertens M(n) and Liouville L(n) functions completely satisfying the stationarity conditions in the broad sense are proved. Keywords: summation arithmetic function, Mertens function, Liouville function, Chebyshev function, number of prime numbers, estimation of the order of growth of the deviation from the mean value of the arithmetic function, standard deviation. DOI: 10.25791/pfim.06.2018.033 Contacts: E-mail: znakvicvolf@mail.ru Pp. 61-65.