A Guide to the Russian Academy of Sciences - University of Texas ...

equations; an upper bound for the number of summands in the Hilbert--Kamke
problem; elementary methods in additive problems on prime number; the study of
the Waring problem and its generalization to non-integer indices; number theory
methods in numerical analysis; the large sieve and its applications.
After I. M. Vinogradov's death, the Department was renamed as the Laboratory of Analytic
Number Theory. At the present time its staff includes: A. A. Karatsuba, Head of
Laboratory; G. I. Arkhipov, S. M. Voronin, V. A. Iskovskikh, and A. I. Pavlov.
The members of the Department have made contributions 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 Vinogradov classical theory of Weil's sums;
problems about the exponent of convergence of special integrals in the Terry
problem and its generalizations were solved; the Hilbert--Kamke problem and its
generalization 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 in finite fields; estimates of short
sums of characters over shifted prime numbers in linear and nonlinear cases were
obtained which are stronger then 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 generalization 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 zetafunction
was proved; on the basis of the Vinogradov method new properties of
solutions of the Cauchy problem for Schrodinger type equations with periodic
initial data were found; 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 large numbers and of rapid calculation of
elementary algebraic functions were found; new quadrature formulae were
constructed; The non-rationality of some classes of three dimensional algebraic
manifolds with zero differential-geometrical invariants was proved, in particular,
the Luroth problem was solved; a birational theory of rational surfaces over an
algebraic non-closed field was developed.
Closely related to the Laboratory research was the work carried out in the Institute by M. P.
Mineev, who studied additive problems with rapidly increasing functions and by A.
I. Pavlov , who worked on lacunar power series, the theory of substitutions with a
given set of cycles and systems of equations in substitutions.
At the present time the actively developed topics at the Laboratory are the theory of multiple
integrals, additive problems, the theory of distribution of prime numbers, the theory
of Riemann zeta-functions and its generalizations, the theory of Dirichlet equations,
the spectral theory of trigonometric series, the theory of birational automorphisms
of three-dimensional Fanc manifolds and defining relations in the Cremona group
of a plane over an algebraic non-closed field.
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