#### Name

Likhareva Yulia Andreevna

#### Scholastic degree

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#### Academic rank

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#### Honorary rank

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#### Organization, job position

Kuban State Technological University

#### Web site url

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## Articles count: 1

The Euler function is very important in number theory
and in Mathematics, however, the range of its values in
the natural numbers has not been written off. The
greatest value of the Euler function reaches on Prime
numbers, furthermore, it is multiplicative. The value of
the Euler function is closely associated with the values
of the Moebius function and the function values of the
sum of the divisors of the given natural number. The
Euler function is linked with systems of public key
encryption. The individual values of the Euler function
behave irregularly because of the irregular distribution
of primes in the natural numbers. This tract is
illustrated in the article with charts; summatory
function for the Euler function and its average value
are more predictable. We prove the formula of
Martinga and, based on it, we study the approximation
accuracy of the average value of the Euler function
with corresponding quadratic polynomial. There is a
new feature associated with the average value of the
Euler function and calculate intervals of its values. We
also introduce the concept of density values of the
Euler function and calculate its value on the interval of
the natural numbers. It can be noted that the results of
the behavior of the Euler function are followed by the
results in the behavior of functions of sums of divisors
of natural numbers and vice versa. We have also given
the results of A.Z.Valfish and A.N.Saltykov on this
subject