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Applied Physics and Mathematics

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ON AN EQUIVALENT FORMULATION OF THE RIEMANN HYPOTHESIS
V.L. VOLFSON

The problem of the distribution of prime numbers has acquired important practical signifi cance recently, in connection with the
development of computer technology and the Internet, because it is directly related to the reliability of so-called cryptographic systems with public key. 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. We consider the equivalent formulation of the Riemann Hypothesis in this paper, which is directly related to the decomposition of natural numbers into prime factors. For any real one performs where is the Mertens function; μ(i) is the Möbius function; i, n are natural numbers; i is natural number having prime divisors only in the fi rst degree. We consider a new Mertens function, which is diff erent from the usual one (without the point 1). This is due to the fact that the value 1 does not have simple dividers. The new Mertens function has exceptional properties. The value of the function is = –1,34164… at the point n = 5, already at the beginning of the natural series, which is much less than the value found earlier in the study of the usual Mertens function, although more 10100 values have been investigated for this. The ratio will be fulfilled with a high probability for the new function of Mertens: , considering how this point was chosen. The equivalent formulation of the Riemann hypothesis follows from this relation.
Keywords: equivalent formulation, Riemann Hypothesis, Mertens function.

Contacts: E-mail: znakvicvolf@mail.ru

Pp. 48-50.

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