#### Name

Sergeev Eduard Alexandrovich

#### Scholastic degree

•

#### Academic rank

â€”

#### Honorary rank

â€”

#### Organization, job position

Kuban State University

#### Web site url

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## Articles count: 6

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

In this article, we present the fundamental theorem of
arithmetic and its role. We consider various rings for
its performance

In 1893, the French mathematician J. Adamar
raised the question: given a matrix of fixed order
with coefficients not exceeding modulo this value,
then what is the maximum modulo value can take
the determinant of this matrix? Adamar fully
decided this question in the case when the
coefficients of the matrix are complex numbers and
put forward the corresponding hypothesis in the
case when the matrix coefficients are real numbers
modulo equal to one. Such matrices satisfying the
Hadamard conjecture were called Hadamard
matrices, their order is four and it is unknown
whether this condition is sufficient for their
existence. The article examines a natural
generalization of the Hadamard matrices over the
field of real numbers, they are there for any order.
This paper proposes an algorithm for the
construction of generalized Hadamard matrices,
and it is illustrated by numerical examples. Also
introduces the concept of constants for the natural
numbers are computed values of this constant for
some natural numbers and shown some
applications of Hadamard constants for estimates
on the top and bottom of the module of the
determinant of this order with arbitrary real
coefficients, and these estimates are in some cases
better than the known estimates of Hadamard. The
results of the article are associated with the results
of the con on the value of determinants of matrices
with real coefficients, not exceeding modulo units

In this article, we discuss various issues related to the
formulas approximating the distribution function of
prime numbers pi(x). This question has occupied many
scholars, but the exact function is well approximated
function pi(x) over the number of positive integers not.
Based on certain hypotheses, we present a new
function s(x) is very well approximated pi(x). The
above article hypotheses are so important that their
numerical validation and refinement for the lengths of
the segments more in 1014 - one of the main areas
related to the problem of approximation of the function
pi(x) throughout the series of natural numbers. After
analyzing the behaviors and constructs many
functions, we are building the basis of the function
s(x), which is well approximates the function pi(x)
throughout the series of natural numbers. We also
present a table of values for x, less or equal 1022 for the
difference of s(x) - pi(x)

The article obtained the explicit form of root polynomials for cyclic polynomials of degree three over fields of characteristic 2. We also give an overview of known results on the root polynomials over arbitrary fields

The article presents the theorem of Chebyshev on the
distribution of primes, considering functions that
approximated prime numbers. We have also
considered a new function, which is quite good for
approximation of prime numbers. A review of the
known results on distribution of prime numbers is
given as well