The history of the specialized boarding school №18 in physics-mathematics at Moscow State University M. V. Lomonosov is described. Listed historical personalities, who stood at the origins of schools for gifted children: academics — A. N. Kolmogorov, I. K. Kikoin, I. G. Petrovsky, M. A. Lavrentiev; Minister of Higher and Secondary Education V. P. Elyutin et al. Shows the historical documents and photographs. Given the personal composition of the first edition (1963–1964 years) specialized boarding school № 18 in physics-mathematics. Described in detail the biography of an outstanding mathematician G. I. Arkhipova — student PMS №18 of the first issue. Briefly described his scientific work at the V. A. Steklov Mathematical Institute Academy of Sciences (1983–2013 years) and pedagogical work at Moscow State University named M. V. Lomonosov. Listed his achievements in the development of the theory of numbers: solving problems posed by Academician IM Vinogradov in "The method of trigonometric sums research on the Hilbert-Kamke problem. GI Research Arkhipova on the Hilbert-Kamke were awarded in 1992 AA awards Markov, Russian Academy of Sciences, which is awarded to mathematicians once every three years. Described textbook "Lectures on mathematical analysis which was written in collaboration with V. A. Sadovnichy and V. N. Chubarikov, withstood six editions. A list of all the published works of G. I. Arkhipova.

Research on the theory of the Riemann zeta function are carried out with great intensity that’s been going on for one and a half centuries, and some parts of the theory became independent scientific directions of modern analytic number theory. An important role among these areas play a theorem about the zeros of the density distribution of the Riemann zeta function in the critical strip. During the last decades, the topic in a large number of scientific articles. She repeatedly touched in scientific monographs and special books on various issues of analytic number theory. Studies of the behavior of the Riemann zeta function ζ(s) in the critical strip essentially based on its proximity segment of the Dirichlet series. The main result of this work is using of the Vinogradov’s method for estimation of ζ(s, k)-Zeta function of the quadratic form K and the growing negative discriminant (−d). In the article is given the use of Vinogradov’s method for estimating ζ(s, k)-Zeta function of the quadratic form K and the growing negative discriminant (−d). Application Vinogradov method for estimating ζ(s, k)-Zeta function of the quadratic form K and the growing negative discriminant (−d). is difficult due to lack of suitable for the purpose of the approximate functional equation. Typically, members of this equation include the factor, which is the value of the character group of divisor classes of the field Q(√ −d) for positive definite quadratic forms of discriminant (−d). This fact is the main obstacle to the effective application of the method of trigonometric sums. C. M. Voronin in his work [1] an approximate functional equation for ζ(s, k) principal term of which represent the initial segment of the Dirichlet series of functions, which are not members of the «twisted» with any character. This allows reducing the question about his assessment to the assessment of the amount of double dzetovoy. The proof is carried out by bringing the zeta-functions of quadratic forms a segment of a Dirichlet series. Also in the article describes the history of the problem behavior of the Riemann zeta function ζ(s) in the critical strip. The basic results of relevance today, shows the application results found.

This paper is a continuation of our research on additive problems of number theory with variables that belong to some special set. We have solved several well–known additive problems such that Ternary Goldbach’s Problem, Hua Loo Keng’s Problem, Lagrange’s Problem,Waring’s Problem. Asymptotic formulas were obtained for these problems with restriction on the set of variables. The main terms of our formulas differ from ones of the corresponding classical problems. In the main terms the series of the form
σk(N, a, b) =X |m|<∞ e2πim(ηN−0,5k(a+b)) sink πm(b − a) πkmk. appear. These series were investigated by the authors. Let η be the irrational algebraic number, a and b are arbitrary real numbers of the interval [0, 1]. There are natural numbers x1, x2, . . . , xk such that a 6 {ηxni } < b. In this paper we evaluate the smallest k for which the equation xn1 + xn2 + . . . + xnk = N is solvable.

Problems related to the distribution of algebraic numbers and points with algebraically conjugate coordinates are a natural generalization of problems connected with estimating of number of integer and rational points in figures and bodies of a Euclidean space. In this paper we consider a problem related to the distribution of special algebraic points α = (α1, α2) with algebraically conjugate coordinates α1 and α2 such that their height and degree are bounded and the absolute values of P′(α1) and P′(α1) where P(t) is a minimal polynomial of α1 and α2 are small. The sphere of application of this points is problems related to Mahler’s classification of numbers [1] proposed in 1932 and Kosma’s classification of numbers [2] proposed some years later. One of this is a question: do Mahler’s T-numbers exist? This question has remained unanswered for nearly 40 years and only in 1970 W. Schmidt [3] showed that the class of T-numbers is not empty and proposed the construction of this numbers. Another problem is a question about difference between Mahler’s and Koksma’s classifications. In 2003 Y. Bugeaud published a paper [4] where he proved that there are exist a numbers with different Mahler’s and Koksma’s characteristics. Special algebraic points α = (α1, α2) considered in this paper are used to prove this results. We consider special algebraic points α = (α1, α2) such that the height of algebraically conjugate numbers α1 and α2 is bounded by Q, their degree is bounded by n and |P′(α1)| ≤ ≤ Q1−v1 , |P′(α2)| ≤ Q1−v2 for 0 < v1, v2 < 1 where P(t) is a minimal polynomial of this numbers. In this paper we obtained the lower and upper bound for the quantity of special algebraic numbers in rectangles with the size of Q−1+v1+v2 .

In this paper, we study the distribution of non-trivial zeros of the Riemann zeta function ζ(s), which are on the critical line ℜs = 1/2. On the half-plane ℜs > 1, the Riemann zeta function is defined by Dirichlet series ζ(s) = X+∞ n=1 n−s, and it can be analytically continued to the whole complex plane except the point s = 1. It is well-known that the non-trivial zeros of the Riemann zeta function are symmetric about the real axis and the line ℜs = 1/2. This line is called critical. In 1859, Riemann conjectured that all non-trivial zeros of the Riemann zeta function lie on the critical line ℜs = 1/2. Hardy was the first to show in 1914 that ζ(1/2 + it) has infinitely many real zeros. In 1942, Selberg obtained lower bound of the correct order of magnitude for the number zeros of the Riemann zeta functions on intervals of critical line [T, T +H],H = T0.5+ε, where ε — an arbitrary small constant. In 1984, A. A. Karatsuba proved Selberg’s result for shorter intervals of critical line [T, T + H],H = T27/82+ε. It is difficult to reduce the length of interval, which was pointed out above. However, if we consider this problem on average, then it was solved by Karatsuba. He proved that almost all intervals of line ℜs = 1/2 of the form [T, T+Xε], where 0 < X0(ε) < X 6 T 6 2X, contain more than c0(ε)Tε ln T zeros of odd orders of the function ζ(1/2+it). In 1988, Kicileva L. V. obtained result of this kind, but for the averaging intervals (X,X + X11/12+ε). In this paper, the length of the averaging interval has reduced. We proved Karatsuba’s result for interval (X,X + X7/8+ε).

Bounded remainder sets are sets with bounded by constant independent of the number of points remainder term of the multidimensional problem of the distribution of linear function fractional parts. These sets were introduced by Hecke and studied by Erd¨os, Kesten, Furstenberg, Petersen, Szusz, Liardet and others. Currently, in one-dimensional case full description of bounded remainder intervals and exact estimates of the remainder term on such intervals are known. Also some more precise results about the remainder term are established. Among these results there are exact formulaes for maximum, minimum and average value of the remainder term, description of the remainder term as piecewise linear function, non-monotonic estimates for the remainder term, estimates of speed of attainment of the remainder term exact boundaries, etc... In the higher dimensional cases only several examples of bounded remainder sets are known. Particularly, in recent years V. G. Zhuravlev, A. V. Shutov, and A. A. Abrosimova introduce a new construction of some families of multidimensional bounded remainder sets based on exchanged toric tilings. For introduced sets we are able not only to prove the boundness of the remainder term but to compute exact values of its minimum, maximum, and average. In the present work we study more subtle property of the remainder term on bounded remainder sets based on exchanged toric tilings: its distribution function. It is proved that the remainder term is uniformly distributed only in one-dimensional case. An algorithm for computation of the normalized distribution function is given. Some structural results about this function are proved. For some two-dimensional families of bounded remainder sets their normalized distribution functions are clealy calculated.

In this paper we consider some hypergeometric functions whose parameters are connected in a special way. Lower estimates of the moduli of linear forms in the values of such functions have been obtained. Usually for the achievement of such estimates one makes use of Siegel’s method; see [1], [2], [3, chapt. 3]. In this method the reasoning begins with the construction by means of Dirichlet principle of the linear approximating form having a sufficiently large order of zero at the origin of coordinates. Employing the system of differential equations, the functions under consideration satisfy, one constructs then a set of forms such that the determinant composed of the coefficients of the forms belonging to this set must not be equal to zero identically. Further steps consist of constructing a set of numerical forms and of proving of the interesting for the researcher assertions: linear independence of the values of the functions under consideration can be proved or corresponding quantitative results can be obtained. By means of Siegel’s method have been proved sufficiently general theorems concerning the arithmetic nature of the values of the generalized hypergeometric functions and in addition to aforementioned linear independence in many cases was established the transcendence and algebraic independence of the values of such functions. But the employment of Dirichlet principle at the first step of reasoning restricts the possibilities of the method. Its direct employment is possible in the case of hypergeometric functions with rational parameters only. It must be taken into consideration also the insufficient accuracy of the quantitative results that can be obtained by this method. As a consequence of these facts some analogue of Siegel’s method has been developed (see [4]) by means of which it became possible in some cases to investigate the arithmetic nature of the values of hypergeometric functions with irrational parameters also. But yet earlier one had begun to apply methods based on effective construction of linear approximating form. By means of such constructions the arithmetic nature of some classic constants was investigated and corresponding quantitative results were obtained, see for example [5, chapt. 1]. Subsequently it turned out that effective methods can be applied also for the investigation of generalized hypergeometric functions. Explicit formulae for the coefficients of the linear approximating forms were obtained. In some cases these formulae make it possible to realize Siegel method scheme also for the hypergeometric functions with irrational parameters. If in (1) polynomial a(x) is equal to unity identically then the results obtained by effective method are of sufficiently general nature and in this case further development of this method meets the obstacles of principal character. In case a(x) ̸≡ 1, however, the possibilities of effective method are not yet exhausted and the latest results can be generalized and improved. In the theorems proved in the present paper new qualitative and quantitative results are obtained for some hypergeometric functions with a(x) = x+α and polynomial b(x) from (1) of special character. The case of irrational parameters is under consideration but the ideas we use will apparently make it possible in the future to obtain new results in case of rational parameters also.

In this paper we consider the algebraic integers of second degree and reducible quadratic monic polynomials with integer coefficients. Let Q > 4 be an integer. Define Ωn(Q, S) to be the number of algebraic integers of degree n and height 6 Q belonging to S ⊆ R. We improve the remainder term of the asymptotic formula for Ω2(Q, I), where I is an arbitrary interval.  Denote by R(Q) the set of reducible monic polynomials of second degree with integer coefficients and height 6 Q. We obtain the formula #R(Q) = 2 XQ k=1 τ (k) + 2Q + hp Q i − 1, where τ (k) is the number of divisors of k. Besides we show that the number of real algebraic integers of second degree and height 6 Q has the asymptotics Ω2(Q,R) = 8Q2 −16 3 Q p Q − 4QlnQ + 8(1 − γ)Q + O p Q , where γ is the Euler constant. It is known that the density function of the distribution of algebraic integers of degree n uniformly tends to the density function of algebraic numbers of degree n−1. We show that for n = 2 the integral of their difference over the real line has nonzero limit as height of numbers tends to infinity.

Estimations of linear sums with Bernoulli polynomial of the first degree are given. If the coefficient of the linear function is a irrational number with the bounded partial quotients, thearithmetical sum has the “squaring” estimation. The Roth’s theorem gives the similar estimation for all algebraic number, but the constants in estimations be nonefficient. New difficulties appears for sums over primes. Their are connected with the consideration of bilinear forms.

Exponential sums of a special type — so-called Kloosterman sums — play key role in the series of number-theoretic problems concerning the distribution of inverse residues in the residual rings of given modulo q. At the same time, in many cases, the estimates of such sums are based on A.Weil’s bound of so-called complete Kloosterman sum of prime modulo. This bound allows one to estimate Kloosterman sums of length N > q0.5+ε for any fixed ε > 0 with power-saving factor. Weil’s bound was proved originally by methods of algebraic geometry. Later, S. A. Stepanov gave an elementary proof of this bound, but this proof was also complete enough. The aim of this paper is to give an elementary proof of Kloosterman sum of length N > q0.5+ε, which also leads to power-saving factor. This proof is based on the trick of “additive shift” of the variable of summation which is widely used in different problems of number theory.

In 1975, Sergei Mikhailovich Voronin discovered the universality of the Riemann zetafunction ζ(s), s = σ + it , on the approximation of a wide class of analytic functions by shifts ζ(s + iτ ), τ ∈ R. Later, it turned out that also some other zeta-functions are universal in the Voronin sense. If τ takes values from a certain descrete set, then the universality is called discrete. In the present paper, the discrete universality of periodic Hurwitz zeta-functions is considered. The periodic Hurwitz zeta-function ζ(s, α; a) is defined by the series with terms am(m + α)−s, where 0 < α ≤ 1 is a fixed number, and a = {am} is a periodic sequence of complex numbers. It is proved that a wide class of analytic functions can be approximated by 
shifts ζ(s+ihkβ1 logβ2 k, α; a) with k = 2, 3, ..., where h > 0 and 0 < β1 < 1, β2 > 0 are fixed numbers, and the set {log(m+α) : m = 0, 1, 2} is linearly independent over the field of rational numbers. It is obtained that the set of such k has a positive lower density. For the proof, properties of uniformly distributed modulo 1 sequences of real numbers are applied.

The paper is devoted to the important problem of number theory: bounded remainder sets. We consider the point orbits on low-dimensional tori. Any starting point generates the orbit under an irrational shift of the torus. The orbit is everywhere dense and uniformly distributed on the torus if the translation vector is irrational. Denote by r(i) a function that gives the number of the orbit points which get some domain T. Then we have the formula r(i) = i vol(T) + δ(i), where δ(i) = o(i) is the remainder. If the boundaries of the remainder are limited by a constant, then T is a bounded remainder set ( BR-set). The article introduces a new BR-sets construction method, it is based on tilings parametric polyhedra. Сonsidered polyhedra are the torus development. Torus development should be to tile into figures, that can be exchanged, and we again obtain our torus development. This figures exchange equivalent shift of the torus. Author have constructed tillings with this property and two-dimensional BR-sets. The considered method gives exact estimates and the average value of the remainder. Also we obtain the optimal BR-sets which have minimal values of the remainder. These BR-sets generate the strong balanced words ( a multi-dimensional analogue of the Sturmian words). The above method is applied to the case of three-dimensional torus in this paper. Also we obtain exact estimates and the average value of the remainder for constructed sets.

This work is dedicated to the study of connection of distribution theory of integral points on the simplest hyperboloid with some hypotheses for Dirichlet L–function. In application of discrete ergodic method (further DEM), developed by U. V. Linnik (see [1, 2]) to the problem of distribution of integral points on hyperboloids x1x3 −x22 = m (as well as and in case of sphere) in formulations of theorems about asymptotically even distribution of integral points some auxiliary prime number p such as that symbol of Legendre 
−m p  = 1. In ergodic theorems and theorems of mixing for integral points the presence of such simple number was natural as it resulted a flow of primitive points used in DEM in conclusion of asymptotic formulae for numbers of integral points on the sphere and on hyperboloid. The receipt (receiving) of residual members in asymptotic formulae for integral points on areas on the sphere and on hyperboloid in frames of usage DEM (see [2, 3]) is of great interest. Studies in this direction for integral points on ellipsoids were carried out by A. V. Malyshev and by author [3] as well as by E. P. Golubeva [4, 5] by means of method of A. U. Vinogradov [6] which are elaboration of dispersions method of U. V. Linnik [7]. It appears that some weakened hypotheses for Dirichlet L–function, directly following from broadened hypotheses of Riman allows to eliminate the mentioned lack. Taking into account that circumstance in combination with that done by A. V. Malyshev and B. M. Shirikov in [8]. There obtained a new proof of key lemma DEM for hyperboloids of both kinds, we give corresponding investigation. In Our work the investigation is done at once for both cases of the simplest hyperboloids and in combination with the use of some hypothesis about the behaviour of Dirichlet L–function and obtain considerable simplification of arguments in results. In connection with our investigation we also note that by the Duke method of modular forms with application of Ivants results [10] we shall obtain asymptotic formulae with absolute residual member for numbers of integral points in areas on the simplest hyperboloid. But in [9] as distinct from our work the distribution of integral points according to classes of deductions according to the given module was not considered. In this connection there appears an interesting problem about transference of Duke’s results [9] to the distribution of integral points of the simplest hyperboloid according to to progressions, i.e. according to classes of deductions.

Classical Poincar´e (θ, p)-inequality on Rn   1 μ(B) ˆ B  f(y) − 1 μ(B) ˆ B f dμ  θ dμ(y)   1/θ . rB   1 μ(B) ˆ B |∇f|p dμ 
   1/p , (rB is the radius of ball B ⊂ Rn) has a self-improvement property, that is (1, p)-inequality, 1 < p < n, implies the «stronger»  q, p)-inequality (Sobolev-Poincar´e), where 1/q = 1/p − 1/n (inequality A . B means that A 6 cB with some inessential constant c). Such effect was investigated in a series of papers for the inequalities of more general type   1 μ(B) ˆ B |f(y) − SBf|θ dμ(y)   1/θ  . η(rB)   1 μ(B) ˆ σB gp dμ   1/p for functions on metric measure spaces. Here f ∈ Lθ loc, g ∈ Lp loc, and SBf is some number depending on the ball B and on the function f, η is some positive increasing function, σ > 1. Usually mean value of the function f on a ball B is chosen as SBf, and the case p > 1 is considered. We investigate self-improvement property for such inequalities on quasimetric measure spaces with doubling condition with parameter γ > 0. Unlike previous papers on this topic we consider the case θ, p > 0. In this case functions are not required to be summable, and we take SBf = I(θ) B f. Here I(θ) B f is the best approximation of the function f in Lθ(B) by constants. We prove that if η(t)t−α increases with some α > 0, then for 0 < p < γ/α and θ > 0 (θ, p)-inequality Poincar´e implies (q, p)-inequality with 1/q > 1/p − γ/α. If p > γ(γ + α)−1 (then the function f is locally integrable) then it implies also (q, p)-inequality with mean value instead of the best approximations I(θ) B f. Also we consider the cases αp = γ and αp > γ. If αp = γ, then (q, p)-inequality with any q > 0 follows from Poincar´e (θ, p)-inequality and moreover some exponential Trudinger type inequality is true. If αp > γ then Poincar´e (θ, p)-inequality implies the inequality |f(x) − f(y)| . η(d(x, y))[d(x, y)]−γ/p . [d(x, y)]α−γ/p for almost all x and y from any fixed ball B (. does depend on B).

Vinogradov’s method of estimation of exponential sums over primes allowed him to solve the number of arithmetic problems with primes. One of them is a problem of distribution of the values of non-principal character on the sequence of shifted primes. In 1938 he proved that if q is an odd prime, (l, q) = 1, χ(a) is non-principal character modulo q, then T(χ) = X p6x χ(p − l) ≪ x1+ε r 1 q + q x + x−1 6  . (IMV ) This estimate is non-trivial when x ≫ q1+ε and an asymptotic formula for the the number of quadratic residues (non-residues) modulo q of the form p − l, p 6 x follows from it. Later in 1953, I. M. Vinogradov obtained a non-trivial estimate of T(χ) when x > q0,75+ε, q is a prime. It was a surprising result. In fact, T(χ) can be represented as a sum over zeroes of correspondent Dirichlet L — function; So a non-trivial estimate of T(χ) is obtained only for x > q1+ε provided that the extended Riemann hypothesis is true. In 1968 A. A. Karatsuba found a method that allowed him to obtain non-trivial estimate of short sums of characters in finite fields with fixed degree. In 1970 using the modification of his technique coupled with Vinogradov’s method he proved that: if q is a prime number, χ is non-principal character modulo q and x > q 1 2+ε, then the following estimate is true T(χ) ≪ xq− 1 1024 ε2 . In 1985 Z. Kh. Rakhmonov generalized the estimate (IMV) for the case of composite modulo and proved: let D is a sufficiently large positive integer, χ is a non-principal character modulo D, χq is primitive character generated by character χ, then
T(χ) 6 x ln5 x r 1 q + q x τ 2(q1) + x−1 6 τ (q1)  , q1 = p\D p ̸ \q 
p. If a character χ coincides with it generating primitive character  q, then the last estimate is non-trivial for x > q(ln q)13. In 2010 г. J. B. Friedlander, K. Gong, I. E. Shparlinski showed that a non-trivial estimateof the sum T(χq) exists for composite q when x — length of the sum, is of smaller order than q. They proved: for a primitive character χq and an arbitrary ε > 0 there exists such δ > 0 that for all x > q 8 9+ε the following estimate holds: T(χq) ≪ xq−δ. In 2013 Z. Kh. Rakhmonov obtained a non-trivial estimate of T(χq) for the composite modulo q and primitive character χq when x > q 5 6+ε. In this paper the theorem about the estimate of the sum  T(χq) is proved for cubefree modulo q. It is non-trivial when x > q 5 6+ε.

I. M. Vinogradov pioneered the study of short exponential sums with primes. For k = 1 using his method of estimating sums with primes, he obtained a non-trivial estimate for sums of the form Sk(α; x, y) = X x−y<n6x Λ(n)e(αnk), α = a q + λ, |λ| 6 1 qτ , 1 6 q 6 τ when exp(c(ln ln x)2) ≪ q ≪ x1/3, y > x2/3+ε, This estimate is based on “Vinogradov sieve” and for k = 1 utilizes estimates of short double exponential sums of the form Jk(α; x, y,M,N) = X M<m62M a(m) X U<n62N x−y<mn6x b(n)e(α(mn)k), where a(m) and b(n) are arbitrary complex-valued functions, M, N are positive integers, N 6 U < 2N, x > x0, y are real numbers. Later, B. Haselgrove, V. Statulyavichus, Pan Cheng-Dong and Pan Cheng-Biao, Zhan Tao obtained a nontrivial estimate for the sum S1(α; x, y), y > xθ, where q was an arbitrary integer, and successfully proved an asymptotic formula for ternary Goldbach problem with almost equal summands satisfying |pi − N/3| 6 H, H = Nθ, respectively when θ = 63 64 + ε, 279 308 + ε, 2 3 + ε, 5 8  + ε. J. Liu and Zhan Tao studied the sum J2(α; x, y,M,N) and obtained a non-trivial estimate for the sum S2(α; x, y) when y > x 11 16+ε. This paper is devoted to obtaining non-trivial estimates for the sum J3(α; x, y,M,N), with a “long” continuous summation over minor arcs.

G. Gr¨atzer’s gives the following example in his monograph «Universal algebra». Let A be a universal algebra (with some family of operations Σ). Let us take an arbitrary set B ⊆ A. For all of the operations f ∈ Σ (let n be the arity of f) let us look how f transformas the elements of Bn. It is not necessary that f(B) ⊆ B, so in the general case B is not a subalgebra of A. But if we define partial operation as mapping from a subset of the set Bn into the set B. then B be a set with a family of partial operations defined on it. Such sets are called partial universal algebras. In our example B will be a partial universal subalgebra of the algebra A, which means the set B will be closed under all of the partial operations of the partial algebra B. So, partial algebras can naturally appear when studying common universal algebras. The concept of congruence of universal algebra can be generalized to the case of partial algebras. It is well-known that the congruences of a partial universal algebra A always from a lattice, and if A be a full algebra (i.e. an algebra) then the lattice of the congruences of A is a sublattice of the lattice of the equivalence relations on A. The congruence lattice of a partial
universal algebra is its important characteristics. For the most important cases of universal algebra some results were obtained which characterize the algebras A without any congruences except the trivial congruences (the equality relation on A and the relation A2). It turned out that in the most cases, when the congruence lattice of a universal algebra is trivial the algebra itself is definitely not trivial. And what can we say about the algebras A whose equivalence relation is, vice versa, contains all of the equivalence relations on A? It turns out, in this case any operation f of the algebra A is either a constant (|f(A)| = 1) or a projection (f(x1, ..., xi, ..., xn) ≡ xi). Kozhukhov I. B. described the semigroups whose equivalence relations are one-sided congruences. It is interesting now to generalize these results to the case of partial algebras. In this paper the partial n-ary groupoids G are studied whose operations f satisfy the following condition: for any elements x1, ..., xk−1, xk+1, ..., xn ∈ G the value of the expression f(x1, ..., xk−1, y, xk+1, ..., xn) is defined for not less that three different elements y ∈ G. It will be proved that if any of the congruence relations on G is a congruence of the partial n-ary groupoid (G, f) then under specific conditions for G the partial operation f is not a constant.

In this paper the exact values of different widths in the space Bq,γ , 1 ≤ q ≤ ∞ with the weight γ for classes W(r) q,a (Φ, μ) were calculated. These classes is consist from functions f, which are analytic in a circle UR := {z : |z| ≤ R} (0 < R ≤ 1) whose n(n ∈ N)-th derivatives by argument f(r) a is belong to the space Bq,γ(1 ≤ q ≤ ∞, 0 < R ≤ 1) and have an averaged modulus of smoothness of second order majorized by function Φ, and everywhere further assumed that the function Φ(t), t > 0 is an arbitrary function that Φ(0) = 0. The exact inequalities between the best polynomial approximation of analytic functions in a unit disk and integrals consisted from averaged modulus of smoothness of second order functions with r-th derivatives order and concrete weight which is flow out from substantial meaning of problem statement. The obtained result is guarantee to calculate the exact values of Bernshtein and Kolmogorov’s widths. Method of approximation which is used for obtaining the estimation from above the Kolmogorov n-width is learn on L. V. Taykov work which earlier is proved for modulus of smoothness of complex polynomials. The special interest is offer the problem about constructing the best linear methods of approximation of classes functions W(r) q,a (Φ, μ) and connected to it the problem in calculating the exact values of Linear and Gelfand n-widths. The founded best linear methods is depend on given number μ ≥ 1 and in particular when μ = 1 is contain the previous proved results. Also showed the explicit form an optimal subspaces given dimension which are implement the values of widths.

The theory of n-ary groups emerged as a generalization of the theory of ordinary (binary) groups. Many definitions of group theory have n-ary analogue in the theory of n-ary groups. For example, n-ary analogs of abelian groups are abelian and semiabelian n-ary group. n-ary group ⟨G, f⟩ is called semiabelian if it is true identity f(x1, x2, . . . , xn−1, xn) = f(xn, x2, . . . , xn−1, x1). If in the n-ary group ⟨G, f⟩ is true identities f(x1, . . . , xn) = f(xσ(1), . . . , xσ(n)) for any permutation σ ∈ Sn, then it is called abelian. There is a close connection between groups and n-ary groups. We note special case of Gluskin-Hosszu Theorem for semiabelian n-ary groups. On any semiabelian n-ary group ⟨G, f⟩ it is possible to define an abelian group ⟨G,+⟩, where a + b = f(a, c, . . . , c, ¯c, b) for c from G. Then for the element d = f(c, . . . , c) and automorphism ϕ(x) = f(c, x, c, . . . , c, ¯c) of group ⟨G,+⟩, is true equalities ϕ(d) = d, ϕn−1(x) = x for any x ∈ G, f(a1, . . . , an) = a1 + ϕ(a2) + . . . + ϕn−2(an−1) + an + d. Group ⟨G,+⟩ is called the retract of n-ary groups ⟨G, f⟩ and denoted by retc⟨G, f⟩. And the opposite is true: in any abelian group⟨G,+⟩ for selected automorphism ϕ and element d with the above conditions are set semiabelian n-ary group ⟨G, f⟩. n-Ary group ⟨G, f⟩ in this case, called (ϕ, d)-derived from the group ⟨G,+⟩ and denoted by derϕ,d⟨G,+⟩. Let ⟨G, f⟩ = derϕ,d⟨G,+⟩ – semiabelian n-ary group. For every automorphism ϕ′ of group ⟨G,+⟩, which is conjugate to the automorphism ϕ, on the group ⟨G,+⟩ we consider the endomorphism μϕ′ (x) = x+ϕ′(x)+. . .+ϕ′n−2(x). Im μϕ′ – image of this endomorphism. Let ϕ′ = θ ◦ϕ◦ θ−1. Then, for each such automorphism θ have coset θ(d)+Im μϕ′ of the subgroup Im μϕ′ . Collection {θ(d) + Im μϕ′ | θ ∈ Aut ⟨G,+⟩} all such cosets we call defining collection of sets for n-ary group ⟨G, f⟩. It is proved that semiabelian n-ary group ⟨G, f⟩ = derϕ,d⟨G,+⟩ и ⟨G, f′⟩ = derψ,q⟨G,+⟩ are isomorphic iff automorphisms ϕ and ψ are conjugate in group of automorphisms of group ⟨G,+⟩ and defining collection of sets for these n-ary groups is equal up to permutation. 
We study the finite semiabelian n-ary groups. It is shown that any semiabelian n-ary group ⟨G, f⟩ of order |G| = pα1 1 pα2 2 . . . pαk k is isomorphic to the direct product ⟨G1, f1⟩ × ⟨G2, f2⟩ × . . . × ⟨Gk, fk⟩ n-ary pi-groups ⟨Gi, fi⟩ of orders |Gi| = pαi i , where pi – distinct primes. This decomposition is uniquely determined. Based on the above decomposition of finite semiabelian n-ary groups into a direct product of primary semiabelian n-ary groups and for its uniqueness, we come to the main assertion about finite semiabelian n-ary groups: Any semiabelian finite n-ary group is isomorphic to the direct product of primary semiabelian n-ary groups. Any two these decompositions have the same number of factors and primary factors in these decompositions on a the same prime number have the same invariants. It is proved the main theorem on the structure of finite abelian n-ary groups: Any finite abelian n-ary group is isomorphic to the direct product of primary abelian semicyclic n -ary groups. Any two these decompositions have the same number of factors of each order and for each prime divisor of the order of n-ary group the primary factors in these decompositions have the same invariants.

Let τk(n) be the number of solutions of the equation x1x2 · · · xk = n in natural numbers x1, x2, . . . , xk. Let Dk(x) = X n6x τk(n). The problem of obtaining of asymptotic formula for Dk(x) is called Dirichlet divisors problem when k = 2, and generalyzed Dirichlet divisors problem when k > 3. This asymptotic formula has the form Dk(x) = xPk−1(log x) + O(xαk+ε), where Pk−1(x) — is the polynomial of the degree k − 1, 0 < αk < 1, ε > 0 — is arbitrary small number. Generalyzed Dirichlet divisor problem has a rich history. In 1849, L. Dirichlet [1] proved , that αk 6 1 − 1 
k , k > 2. In 1903, G. Voronoi [2] αk 6 1 − 1 k + 1 , k > 2. (see also [3]) In 1922, G. Hardy and J. Littlewood [4] proved that αk 6 1 − 3 k + 2 , k > 4. In 1979, D. R. Heath-Brown [5] proved that 
αk 6 1 − 3 k , k > 8. In 1972, A. A. Karatsuba got a remarkable result [6]. His uniform estimate of the remainder term has the form O(x1− c k2/3 (c1 log x)k), where c > 0, c1 > 0 — are  bsolute constants. Let N0 — be a set of natural numbers whose binary expansions have even number of ones. In 1991, the autor [8] solved Dirichlet divisors problem and got the formula X n6X n∈N0 τ (n) = 1 2 X n6X τ (n) + O(Xω ln2 X),  where τ (n) — the number of divisors n, ω = 1 2

Vladimir Igorevich Parusnikov died on August 22, 2015 after a long and serious illness. He was born in Moscow on January 21, 1957. Vladimir Igorevich graduated from the Department of Mechanics and Mathematics of the Lomonosov Moscow State University, finished his postgraduate studies in 1982 and he defended his thesis in 1983 in there. At the Keldysh Institute of Applied Mathematics he worked since 1982 as a junior researcher, and since 1996 — as a senior researcher. He has published more than 45 scientific papers on functional and numerical continued fractions and their generalizations.V. I. Parusnikov was kind, honest, responsible man and a talented mathematician. His passing is a great loss for the KIAM and for science in general. He was buried at Khovanskoye cemetery in Moscow. We provide a survey of his mathematical work. First, he studied the generalized functional continued fractions and got rather strong results. The last twenty years he, together with A. D. Bruno, was looking for a multi-dimensional generalization of the continued fraction giving the best Diophantine approximations and the periods in the algebraic case. Such a generalization was finally found. The first Section of the paper was written by A. I. Aptekarev, the second Section was written by A. D. Bruno, the list of scientific publications of V. I. Parusnikov was prepared by A. B. Batkhin.