The invited talk presented at the seminar of Prof. B. S. Kashin and Prof. S. V. Konyagin at the Faculty of Mechanics and Mathematics of Moscow Lomonosow University at the November 9, 2006.

 

The invited talk presented at the seminar of Prof. P.M. Gruber at the Chair of Mathematical Analysis of the Department of Mathematics of Vienna University of Technology at the June 13, 1994.

The authors set themselves two main objectives: to characterize the main stages of the life of the outstanding Russian mathematician, Honored Scientist of Russia, Professor of Moscow State University named after M. V. Lomonosov, head of the department of number theory Steklov Mathematical Institute, Doctor of Physical and Mathematical Sciences Anatolii Alexeevich Karatsuba and give a brief analysis of his scientific work, which has had a significant impact on the development of analytic number theory. Sufficiently detailed description of the research of Professor A. A. Karatsuba and his disciples on the analytic theory of numbers, which are allocated following the Karatsuba, three main areas: 1) trigonometric sums and trigonometric integrals; 2) the Riemann zeta function; 3) Dirichlet characters. A. A. Karatsuba, being a disciple of Professor N. M. Korobov, led the scientific schools and seminars on analytic number theory at Moscow State University named after M. V. Lomonosov. Among his many students defended their dissertations, with seven of them later became Doctor of Physical and Mathematical Sciences. Anatoly Alekseevich has published 158 scientific papers, including 4 monographs and a classic textbook on analytic number theory, was the translator of a number of fundamental scientific monographs. He was a member of the editorial board of the journal "Mathematical notes"and a member of the program committees of several international conferences on algebra and number theory.

Lyapunov matrix equations and their generalizations — linear matrix Sylvester equation widely used in the theory of stability of motion, control theory, as well as the solution of differential Riccati and Bernoulli equations, partial differential equations and signal processing. If the structure of the general solution of the homogeneous part of the Lyapunov equation is well studied, the solution of the inhomogeneous equation Sylvester and, in particular, the Lyapunov equation is quite cumbersome. By using the theory of generalized inverse operators, A. A. Boichuk and S. A. Krivosheya establish a criterion of the solvability of the Lyapunov-type matrix equations AX - XB = D and X - AXB = D and investigate the structure of the set of their solutions. The article A. A. Boichuk and S. A. Krivosheya based on pseudo-inverse linear matrix operator L, corresponding to the homogeneous part of the Lyapunov type equation. The article suggests the solvability conditions, as well as a scheme for constructing a particular solution of the inhomogeneous generalized equation Sylvester based on pseudo-inverse linear matrix operator corresponding to the homogeneous part of the linear matrix generalized Sylvester equation. Using the technique of Moore-Penrose pseudo inverse matrices, we suggest an algorithm for finding a family of linearly independent solutions of the inhomogeneous generalized equation Sylvester and, in particular, the Lyapunov equation in general case when the linear matrix operator L, corresponding to the homogeneous part of the linear generalized matrix Sylvester equation, has no inverse. We find an expression for family of linearly independent solutions of the inhomogeneous generalized equation Sylvester and, in particular, the Lyapunov equation in terms of projectors and Moore-Penrose pseudo inverse matrices. This result is a generalization of the result article A. A. Boichuk and S. A. Krivosheya to the case of linear generalized matrix Sylvester equation. The suggested the solvability conditions and formula for constructing a particular solution of the inhomogeneous generalized equation Sylvester is illustrated by an examples.

A well founded way of researching the linear algebra is the study of it using the identities, consequences of which is the identity of nilpotent. We know the Nagata-Higman’s theorem that says that associative algebra with nil condition of limited index over a field of zero characteristic is nilpotent. It is well known the result of E.I.Zel’manov about nilpotent algebra with Engel identity. A set of linear algebras where a fixed set of identities takes place, following A.I. Maltsev, is called a variety. The variety is called almost nilpotent if it is not nilpotent, but each its own subvariety is nilpotent. Recently has been studied the growth of the variety. There is a variety of polynomial, exponential, overexponential growth, a variety with intermediate between polynomial and exponential growth. A variety has subexponential growth if it has polynomial or intermediate growth. This article is a review and description of almost nilpotent varieties in different classes of linear algebras over a field of zero characteristic. One part of the article is devoted to the case of classical linear algebras. Here we present the only associative almost nilpotent variety, it is the variety of all associative and commutative algebras. In the case of Lie algebras the almost nilpotent variety is the variety of all metabelian Lie algebras. In the case of Leibniz algebras we prove that there are only two examples of almost nilpotent varieties. All presented almost nilpotent varieties in this section have polynomial growth. In general case it was found that there are rather exotic examples of almost nilpotent varieties. In this work we describe properties of almost nilpotent variety of exponent 2, and also the existence of a discrete series of almost nilpotent varieties of different integer exponents is proved. The last section of the article is devoted to varieties with subexponential growth. Here we introduce almost nilpotent varieties for left-nilpotent varieties of index two, commutative metabelian and anticommutative metabelian varieties. As result we found that each of these classes of varieties contain exactly two almost nilpotent varieties.

 

In January, 2014, the I’st one-day international “Conference to the Memory of A.A. Karatsuba on Number Theory and Applications” took place in Steklov Mathematical Institute of Russian Academy of sciences. The aims of this conference were presentation of new and important results in different branches of number theory (especially in branches connected with works of A. A. Karatsuba), the exchange by new number-theoretical ideas and insight with new methods and tendencies in number theory. The 2’nd Conference was organized by Steklov Mathematical Institute of Russian Academy of sciences together with Moscow State university in January, 2015. The present paper contains wide annotations of reports of 2’nd Conference.

 

 

Let n ∈ N be fixed, Q > 1 be some natural parameter, and Pn(Q) denote the set of integer polynomials of degree n and height of at most Q. Given a polynomial P(x) = anx n + · · · + a0 ∈ Z[x] of degree n, the discriminant of P(x) is defined by D(P) = a 2n−2 n ∏ 16i< |D(P)| 6 Q 2n−2−2v . The first results for the estimate of the number of polynomials with given discriminants were received by H. Davenport in 1961, which were crucial to the solving of the problem of Mahler. In this paper for the first time we obtain the exact upper and lower bounds for #P3(Q, v) with the additional condition on the distribution of the roots of the polynomials. It is interesting that the value of #Pn(Q, v) has the largest value when all the roots of polynomials are close to each other. If there are only k, 2 6 k < n, close roots to each other then the value of #Pn(Q, v) will be less.

The article gives an expanded version of the report, the author of Made in January 30, 2015 in Moscow at an international conference, dedicated to the memory of Professor A. A. Karatsuba, held at the Mathematical Institute. Russian Academy of Sciences and Moscow State University named after M. V. Lomonosov. The report sets out the facts from the history of the theory of hyperbolic zeta function, provides definitions and notation. The main content of the report was focused discussion of actual problems of the theory of hyperbolic zeta function of lattices. Identified the following promising areas of current research: 1. The problem of the correct order of decreasing hyperbolic zeta function in α → ∞; 2. The problem of existence of analytic continuation in the left half-plane α = σ + it(σ 6 1) hyperbolic zeta function of lattices ζH(Λ|α); 3. Analytic continuation in the case of lattices S. M. Voronin Λ(F, q); 4. Analytic continuation in the case of joint lattice approximations; 5. Analytic continuation in the case of algebraic lattices Λ(t, F) = tΛ(F); 6. Analytic continuation in the case of an arbitrary lattice Λ.; 7. The problem behavior hyperbolic zeta function of lattices ζH(Λ|α) in the critical strip; 8. The problem of values of trigonometric sums grids. As a promising method for investigating these problems has been allocated an approach based on the study of the possibility of passing to the limit by a convergent sequence of Cartesian grids.

 

Till recently, even for quadratic algebraic numbers, it was unknown, how frequently do algebraic numbers appear in an arbitrary interval depending on its position and length. Let An be the set of algebraic numbers of n-th degree, and let H(α) be the naive height of α that equals to the naive height of its minimal polynomial by definition. The above problem comes to the study of the following function: Φn(Q, x) := # {α ∈ An ∩ R : H(α) 6 Q, α < x} . The exact asymptotics of Φn(Q, x) as Q → +∞ was recently obtained by the author. There, in fact, the density function of real algebraic numbers was correctly defined and explicitly described. In the paper, we discuss the results on the distribution of real algebraic numbers. For n = 2, we improve an estimate of a remainder term in the asymptotics of Φ2(Q, x), and obtain the following formula: Φ2(Q, +∞) = λ Q3 − κ Q2 ln Q + O(Q 2 ), where λ and κ are effective constants.

In 1975, S. M. Voronin obtained the universality of Dirichlet L-functions L(s, χ), s = σ +it. This means that, for every compact K of the strip {s ∈ C : 1 2 < σ < 1}, every continuous non-vanishing function on K which is analytic in the interior of K can be approximated uniformly on K by shifts L(s+iτ, χ), τ ∈ R. Also, S. M. Voronin investigating the functional independence of Dirichlet L-functions obtained the joint universality. In this case, a collection of analytic functions is approximated simultaneously by shifts L(s + iτ, χ1), . . . , L(s + iτ, χr), where χ1, . . . , χr are pairwise non-equivalent Dirichlet characters. The above universality is of continuous type. Also, a joint discrete universality for Dirichlet L-functions is known. In this case, a collection of analytic functions is approximated by discrete shifts L(s + ikh, χ1), . . . , L(s + ikh, χr), where h > 0 is a fixed number and k ∈ N0 = N ∪ {0}, and was proposed by B. Bagchi in 1981. For joint discrete universality of Dirichlet L-functions, a more general setting is possible. In [3], the approximation by shifts L(s + ikh1, χ1), . . . , L(s+ikhr, χr) with different h1 > 0, . . . , hr > 0 was considered. This paper is devoted to approximation by shifts L(s + ikh1, χ1), . . . , L(s + ikhr1 , χr1 ), L(s + ikh, χr1+1), . . . , L(s + ikh, χr), with different h1, . . . , hr1 , h. For this, the linear independence over Q of the set L(h1, . . . , hr1 , h; π) = { (h1 log p : p ∈ P), . . . ,(hr1 log p : p ∈ P), (h log p : p ∈ P); π } , where P denotes the set of all prime numbers, is applied.

In 1975, a Russian mathematician S. M. Voronin discovered the universality property of the Riemann zeta-function ζ(s), s = σ+it. Roughly speaking, this means that analytic functions from a wide class can be approximated uniformly on compact subsets of the strip {s ∈ C : 1/2 < σ < 1} by shifts ζ(s + iτ ), τ ∈ R. Later, it turned out that other classical zeta and L-functions are also universal in the Voronin sense. Moreover, some zeta and L-functions have a joint universality property. In this case, a given collection of analytic functions is approximated simultaneously by shifts of zeta and L-functions. In the paper, we present our extended report given at the Conference dedicated to the memory of the famous number theorist Professor A. A. Karacuba. The paper contains the basic universality results on the so-called mixed joint universality initiated by H. Mishou who in 2007 obtained the joint universality for the Riemann zeta and Hurwitz zeta-functions. In a wide sense the mixed joint universality is understood as a joint universality for zeta and L-functions having and having no Euler product. In 1989, A. Selber introduced a famous class S of Dirichlet series satisfying certain natural hypotheses including the Euler product. Periodic Hurwitz zetafunctions are a generalization of classical Hurwitz zeta-functions, and have no Euler product. In the paper, a new result on mixed joint universality for L-functions from the Selberg clas and periodic Hurwitz zeta-functions is presented. For the proof a probabilistic method can be applied.

An asymptotic formula is obtained in generalized Estermann’s ternary problem for noninteger powers with almost equal summands on the representations of a sufficiently large natural number as a sum of two primes and integer part of noninteger power of a natural number.

Arithmetic properties of series of the form ∑∞ n=0 an · n! with an ∈ Z are studied. The concept of infinite algebraic independence polyadic numbers. A theorem on the algebraic independence polyadic infinite number of class F (Q, C1, C2, C3, d0), if they are connected by a system of linear differential equations of a certain kind.

 

Memories Alexei Dmitrievich Nadёzhina allow readers to get to know the identity of the remarkable scientist and man — Anatolii Alexeevich Karatsuba. They allow you to plunge into the unique world of mountain climbing, which have a lasting effect on the formation of personality.

 

The article introduces the known discrete transformation – a mirror reflection (in other words – mirror transformation) from a new point of view. The mirror symmetry leads to the preservation of P-parity (spatial parity) in physical phenomena. The mirror symmetry has not been questioned – till recently – the reflection in the mirror interchanges the right and the left, in other respects original object and its reflection remain completely identical. In this work we show that this situation is apparent at first glance, but in general does not correspond to reality. In most cases, the real experimental situation is described by the vectors, and in most cases a combination of true vectors (polar vectors) and pseudovectors (axial vectors) takes place. The vectors of these two types behave differently in the mirror, while the overall reflection in the mirror is assymetrical to the initial object. This situation is applied both to a single mirror transformation and to the spatial inversion, which is equivalent to the successive reflection in three mutually perpendicular mirrors. Both of these versions are considered in detail in this paper. The discovery of P-parity nonconservation in 1956 caused a shock in the physical circles. An attempt of introducing the combined CP-parity instead of P-parity was made. It was not successful, as experiment showed CP-parity is not conserved in the decay of kaons. The essence of CP-parity nonconservation (for more than half century) has had no satisfactory conventional solutions. We believe that the given article gives a solution and it is connected with asymmetry of mirror reflection. Furthermore we believe that P-parity nonconservation is possible not only in the physical processes caused by the weak interaction, but also in the processes related to other types of interactions – electromagnetic, strong. Thus, this paper introduces a new aspect of the relationship of space and physical phenomena.

The article is devoted to the 75th anniversary of Professor Vladimir N. Bezverkhnii. The authors made an attempt to describe the work of V. N. Bezverkhnii in the area of combinatorial group theory. Also some personal reminiscences are included. V. N. Bezverkhnii, who is a former student of Martin Greendlinger, plays major role in scientific school “Algorithmic problems in theory of groups and semigroups”. Seven his students received their PhD degrees under his supervision, and one of them became a Doctor of Science.

 

The authors set themselves two main objectives: to characterize the main stages of the life of Associate Professor, Head of the Department of Physics and Mathematics Faculty of Tula State Pedagogical Institute name of Leo Tolstoy Vladimir Dmitrievich Podsypanin and give a brief analysis of its research and teaching, a significant impact on the development of the Tula scientific-theoretical school. Highlights the research assistant professor V. D. Podsypanin and his students on algebraic number theory and Diophantine analysis. V. D. Podsypanin, being a disciple of the corresponding member of the Academy of Sciences of the USSR, Professor D. K. Faddeev, led the scientific schools and seminars on the theory of numbers in TGPI name of Leo Tolstoy. Among his many students defended their dissertations. Vladimir Dmitrievich Podsypanin have deep, meaningful scientific work. With the bulk of his work was not published until 42 years after his death. He was active in the Abstract Journal Mathematics of the founding of the magazine until his death.

 

The two major goals of the authors are to describe the principal stages of the life of Professor of the Moscow State Pedagogical University Vassiliy Ilyich Nechaev and to give a brief analysis of his scientific and educational activities that has been very influential in the development of both number theory and methods of teaching mathematics at colleges for teacher education. Particular attention is given to the research of Professor V. I. Nechaev and his disciples in analytic number theory and its applications. V. I. Nechaev, being a student of Professor M. K. Grebencha, was a leading figure at the scientific school and the organizer of the analytic number theory research seminar at the Moscow State Pedagogical University. Many of his students received their PhD degrees and later one of them awarded a habilitation in pedagogical sciences. Vassiliy Ilyich Nechaev has published a large number of scientific and methodical works, was a translator of several fundamental scientific monographs. He served as a member of the editorial board of the journal "Mathematical Notes" and as a member of the program committees of several international conferences on algebra and number theory held at the Lev Tolstoy Tula State Pedagogical University.