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Applied Physics and Mathematics

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AN INVESTIGATION OF THE ASYMPTOTIC BEHAVIOR OF THE MERTENS FUNCTION
V.L. VOLFSON

The problem of reliable information transmission became of primary importance with the development of the
Internet. Now we are successfully using public key encryption with the RSA algorithm for increasing reliability. The
RSA algorithm is based on the computational complexity of the prime factorization of large natural numbers. The paper
deals with the Mertens function, which is directly related to the expansion of natural numbers by prime factors. The
Mertens function has the form: , where μ(i) is Mobius function. In the case that a natural number i has an
even number of prime divisors of the fi rst degree, then the value is μ(i) = 1. If the natural number i has an odd number
of prime divisors of the fi rst degree, then the value is μ(i) = –1. If the natural number i has prime divisors of degree
higher than the fi rst, then the value is μ(i) = 0. In the paper we prove the relation: where F(y)
is the limit function of the distribution of the Mobius function. The relation is proved:
where M(n) is the Mertens function, G(y) is the function of the standard normal distribution, and It is shown
that the law of the iterated logarithm is satisfi ed for a random variable having a limiting distribution, as the Mertens
function. Therefore, the hypothesis (1.6) and, consequently, the hypothesis (1.5) are fulfi lled for the Mertens function,
almost everywhere
Keywords: Mobius function, Martens function, growth order, probability space, random variable, limit distribution,
normal distribution, law of the iterated logarithm.

Contacts: E-mail: znakvicvolf@mail.ru

Pp. 46-50.

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