Applied Physics and Mathematics Annotation << Back ASYMPTOTICS OF SUMMATION FUNCTIONS AND THE RIEMANN CONJECTURE V.L. VOLFSON Recently, with the development of computer technology and the Internet, the problem of the distribution of prime numbers has acquired 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. In this paper, we investigate the problem of the asymptotics of summation functions. We will show that it is directly related to the problem of the distribution of primes. In the paper, it will be proved that the best asymptotic estimates for summation functions will be provided that the Riemann conjecture is satisfied. Keywords: arithmetic function, summation function, asymptotic upper bound, Riemann conjecture, Perron’s formula, Mertens function, Chebyshev function, equivalent formulation of the Riemann conjecture. DOI: 10.25791/pfim.06.2020.1187 Pp. 53-56.