Department of Number Theory

 Staff
 History of the Department
 Main results

The Department of Number Theory was organized in 1934. Academician I. M. Vinogradov was the Head of the Department in 1934–1983. Since 1983 Prof. A. A. Karatsuba is the Head of the Department. At different times, K. K. Mardzhanishvili, A. O. Gelfond, B. I. Segal, L. G. Shnirel'man, N. M. Korobov, L. P. Postnikova, N. V. Kuznetsov, S. A. Stepanov, A. I. Vinogradov, A. G. Postnikov, K. I. Oskolkov, S. M. Voronin, A. I. Pavlov, I. Yu. Fedorov, and M. E. Tchanga worked in the Department.

The most brilliant research achievements of the members of the Department include:

• a new method of estimates of H. Weyl's sums and its applications in number theory;
• an asymptotic formula for the number of representations of an odd integer by a sum of three prime numbers and, as corollary of this formula, the solution of the Goldbach problem;
• the theory of trigonometric sums with prime numbers;
• the solution of the 7th Hilbert problem on transcendency of logarithms of algebraic numbers;
• rational approximations of linear forms of algebraic numbers and Diophantine equations;
• the upper bound for the number of summands in the Hilbert–Kamke problem;
• elementary methods in additive problems with prime numbers;
• the Waring problem and its generalizations to non-integer exponents;
• the number theory methods in numerical analysis;
• the large sieve and its applications.

Research projects
At the present time the actively developed topics at the Department are

• the theory of multiple trigonometric sums; the theory of trigonometric integrals;
• additive problems;
• the theory of distribution of prime numbers;
• the theory of the Riemann zeta-function and its generalizations;
• the theory of Dirichlet's characters;
• the theory of Diophantine equations;
• the theory of infinite series;
• the Minimal Model Program and its applications to birational classification of higher-dimensional algebraic varieties;
• birational rigidity of higher-dimensional Fano varieties and Fano fiber spaces;
• birational classification of Mori fiber spaces.

The members of the Department have made contribution to all main directions of analytic number theory as well as to some directions of applied mathematics, function theory, and algebraic geometry. In particular,

• a local method of trigonometric sums was suggested which was used to construct a theory of multiple trigonometric sums similar to the classical Vinogradov's theory of Weyl's sums;
• o problems about the exponent of convergence of singular integrals in the Tarry problem and its generalizations were solved;
• the Hilbert–Kamke problem and its generalizations to the multiple case were solved;
• it was proved that strong forms of the Artin hypothesis on the number of variable forms or systems of forms, representing non-trivial zero in local fields are false;
• a method of estimation of short sums of characters with modules equal to a power of a fixed prime number was discovered;
• new elementary methods were developed in the theory of distribution of prime numbers and in the theory of equations over a finite field;
• estimates of short sums of characters over shifted prime numbers in linear and non-linear case were obtained which are stronger than the results implied by the extended Riemann hypothesis;
• the universality of the Riemann zeta-function and its generalizations was proved;
• a new method of obtaining explicit formulae in additive problems of number theory was suggested;
• a strong version of the Hilbert problem on differential independence of the Riemann zeta-function and its generalizations was proved;
• the A. Selberg hypothesis on zeros of the Riemann zeta-function on short intervals of the critical line was proved;
• a theorem about the `exclusiveness' of the critical line for zeros of the Davenport–Heilbronn function and the Epstein zeta-function was proved;
• on the basis of the Vinogradov method new properties of solutions of the Cauchy problem for Schroedinger type equations with periodic initial data were found and, in particular, a `quantum chaos' was discovered;
• local and global properties of sums of trigonometric series with real algebraic polynomials in the index of imaginary exponent were studied;
• algorithms of rapid multiplications of multi-digit numbers and of rapid calculation of elementary algebraic functions were found;
• new quadrature formulae were constructed;
• the three dimensional Luroth problem was solved;
• a theory of rational surfaces over an algebraically non-closed field was developed and defining relations in the Cremona group of plane over an algebraically non-closed field were described;
• the concept of birational rigidity, which is now one of the crucial concepts of higher dimensional birational geometry, was introduced and studied, a birational rigidity was proved for the main classes of higher dimensional Fano varieties and large classes of Fano fiber spaces.