


Department of Number Theory
 Seminars 
History of the Department 
Research Directions 
Main results 
Publications 


Staff

Konyagin Sergei Vladimirovich Doctor Phys.Math. Sci., Professor, Academician of RAS, Head of Department, Chief Scientific Researcher
office: 520; tel.: +7 (495) 984 81 41 * 3665; email: konyagin23@gmail.com, konyagin@miras.ru Principal fields of research:
Trigonometric series. Polynomials. Best approximation. Character sums.


Balkanova Olga Germanovna PhD, Scientific Researcher
office: 509; tel.: +7 (495) 984 81 41 * 36 07; email: balkanova@miras.ru Principal fields of research:
Analytic number theory, Lfunctions.


Frolenkov Dmitrii Andreevich Candidate Phys.Math. Sci., Senior Scientific Researcher
office: 509; tel.: +7 (495) 984 81 41 * 36 07; email: frolenkov@miras.ru Principal fields of research:
Analytic number theory. Automorphic functions. Kloosterman sums. $L$functions. Continued fractions.


Gabdullin Mikhail Rashidovich Candidate Phys.Math. Sci., Scientific Researcher
office: 520; tel.: +7 (495) 984 81 41 * 3665; email: gabdullin@miras.ru Principal fields of research:
analytic number theory, character sums, quadratic residues.


Korolev Maxim Aleksandrovich Doctor Phys.Math. Sci., Science Deputy Director, Leading Scientific Researcher
office: 324, 526; tel.: +7 (499) 941 03 62, +7 (495) 984 81 41 * 37 32; email: korolevma@miras.ru Principal fields of research:
Incomplete Kloosterman sums, argument of Riemann's zetafunction, power residues, the average number of power residues, the problem of Lehmer–Landau.


Rezvyakova Irina Sergeevna Candidate Phys.Math. Sci., Senior Scientific Researcher
office: 526; tel.: +7 (495) 984 81 41 * 37 32; email: rezvyakova@miras.ru Principal fields of research:
Analytic number theory.


Shkredov Il'ya Dmitrievich Doctor Phys.Math. Sci., Corresponding Member of RAS, Chief Scientific Researcher
office: 509; tel.: +7 (495) 984 81 41 * 36 07; email: ilya.shkredov@gmail.com, ishkredov@rambler.ru Personal page: https://homepage.miras.ru/~ishkredov/
Principal fields of research:
Additive combinatorics, number theory, ergodic number theory.



Arkhipov Gennadii Ivanovich (12.12.1945 – 14.03.2013) Doctor Phys.Math. Sci.
Principal fields of research:
Number theory, mathematical analysis.


Iskovskikh Vasilii Alekseevich (1.07.1939 – 4.01.2009) Doctor Phys.Math. Sci., Corresponding Member of RAS
Principal fields of research:
Rationality problem, birational rigidity, birational classification.


Karatsuba Anatolii Alekseevich (31.01.1937 – 28.09.2008) Doctor Phys.Math. Sci.
Personal page: https://homepage.miras.ru/~karatsuba
Principal fields of research:
Analytic number theory and mathematical cybernetics.



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Seminars

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History of the Department
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.
In 2010 the Department of Number theory was merged with the Department
of Algebra. Aleksei N. Parshin, corresponding member of the Russian
Academy of Sciences, was appointed the head of the new Department of
Algebra and Number theory.
In 2016 the Department of Number theory has been reestablished as as
an independent unit of the institute. Head of the Department is S.V.
Konyagin, corresponding member of the Russian Academy of Sciences.
At different times, G. I. Arkhipov, 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 noninteger exponents;
 the number theory methods in numerical analysis;
 the large sieve and its applications.

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Research Directions
Nowadays the number theory research is concentrated on
additive combinatorics:
the theory of sum products, estimates on the cardinality of sets having
no solutions of linear equations;
analytic number theory:
distribution of prime numbers, the theory of the Riemann zeta function
and its generalizations,
Gram&39;s law, the theory of Dirichlet characters, incomplete Kloosterman
sums, power residues, different additive problems, the theory of
automorphic $L$series, convolution formulas.

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Main results
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 nontrivial 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 nonlinear case were obtained which are stronger
than the results implied by the extended Riemann hypothesis;
 the universality of the Riemann zetafunction 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
zetafunction and its generalizations was proved;
 the A. Selberg hypothesis on zeros of the Riemann zetafunction 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 zetafunction 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 multidigit 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
nonclosed field was developed and defining relations in the Cremona group of plane
over an algebraically nonclosed 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.

List of publications

