Anatoly Alexeyevich Karatsuba (his first name often spelled Anatolii) (Russian: Анато́лий Алексе́евич Карацу́ба; Grozny, Soviet Union, 31 January 1937 – Moscow, Russia, 28 September 2008[1]) was a Russian mathematician working in the field of analytic number theory, p-adic numbers and Dirichlet series.

For most of his student and professional life he was associated with the Faculty of Mechanics and Mathematics of Moscow State University, defending a D.Sc.

[2] He later held a position at the Steklov Institute of Mathematics of the Academy of Sciences.

[6] Up to this day (2011) this result of Karatsuba that later acquired the title "the Moore-Karatsuba theorem", remains the only precise (the only precise non-linear order of the estimate) non-linear result both in the automata theory and in the similar problems of the theory of complexity of computations.

-series modulo a power of a prime number, to the asymptotic formula for the number of Waring congruence of the form to a solution of the problem of distribution of fractional parts of a polynomial with integer coefficients modulo

-adic method of A.A. Karatsuba is the transition from incomplete systems of equations to complete ones at the expense of the local

This led to an essentially new estimate of trigonometric sums and a new mean value theorem for such systems of equations.

and, conversely, the techniques of estimating those parameters in terms of the measure of this set in the real and

This side of Karatsuba's method manifested itself especially clear in estimating trigonometric integrals, which led to the solution of the problem of Hua Luogeng.

Chubarikov obtained a complete solution[15] of the Hua Luogeng problem of finding the exponent of convergency of the integral: where

Karatsuba and his students obtained a series of new results connected with the multi-dimensional analog of the Tarry problem.

This result, being not a final one, generated a new area in the theory of trigonometric integrals, connected with improving the bounds of the exponent of convergency

-adic representation of zero by a form of arbitrary degree d. Artin initially conjectured a result, which would now be described as the p-adic field being a C2 field; in other words non-trivial representation of zero would occur if the number of variables was at least d2.

Karatsuba showed that, in order to have a non-trivial representation of zero by a form, the number of variables should grow faster than polynomially in the degree d; this number in fact should have an almost exponential growth, depending on the degree.

Up to the early 1990s, the estimates of this type were known, mainly, for sums in which the number of summands was higher than

increases to infinity (this case was studied by A. G. Postnikov by means of the method of Vinogradov).

Karatsuba's method makes it possible to estimate Kloosterman sums where the number of summands does not exceed and in some cases even where

Various aspects of the method of Karatsuba have found applications in the following problems of analytic number theory:

[28] The estimates of Atle Selberg and Karatsuba can not be improved in respect of the order of growth as

Karatsuba developed a new method [30][31] of investigating zeros of functions which can be represented as linear combinations of Dirichlet

Similar results were obtained by Karatsuba also for linear combinations containing arbitrary (finite) number of summands; the degree exponent

To Karatsuba belongs a new breakthrough result [32] in the multi-dimensional problem of Dirichlet divisors, which is connected with finding the number

has no zeros in the region Karatsuba found [33](2000) the backward relation of estimates of the values

In the end of the sixties Karatsuba, estimating short sums of Dirichlet characters, developed [37] a new method, making it possible to obtain non-trivial estimates of short sums of characters in finite fields.

Karatsuba developed a number of new tools, which, combined with the Vinogradov method of estimating sums with prime numbers, enabled him to obtain in 1970 [38] an estimate of the sum of values of a non-principal character modulo a prime

In 1971 speaking at the International conference on number theory on the occasion of the 80th birthday of Ivan Matveyevich Vinogradov, Academician Yuri Linnik noted the following: «Of a great importance are the investigations carried out by Vinogradov in the area of asymptotics of Dirichlet character on shifted primes

runs through a sequence of primes in an arithmetic progression, the increment of which may grow together with the modulus

The most characteristic example of that kind is the following claim which is applied in solving a wide class of problems, connected with summing up values of Dirichlet characters.

Karatsuba applied his method also to the problems of distribution of power residues (non-residues) in the sequences of shifted primes

Applying his ATS theorem and some other number-theoretic approaches, he obtained new results[44] in the Jaynes–Cummings model in quantum optics.