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Sergeev Eduard Alexandrovich

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## Articles count: 6

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