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.

Anatoly Alexeyevich Karatsuba
Born(1937-01-31)31 January 1937
Died28 September 2008(2008-09-28) (aged 71)
NationalityRussian
Alma materMoscow State University
Scientific career
FieldsMathematician
Doctoral advisorN. M. Korobov

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. there entitled "The method of trigonometric sums and intermediate value theorems" in 1966.[2] He later held a position at the Steklov Institute of Mathematics of the Academy of Sciences.[2]

His textbook Foundations of Analytic Number Theory went to two editions, 1975 and 1983.[2]

The Karatsuba algorithm is the earliest known divide and conquer algorithm for multiplication and lives on as a special case of its direct generalization, the Toom–Cook algorithm.[3]

The main research works of Anatoly Karatsuba were published in more than 160 research papers and monographs.[4]

His daughter, Yekaterina Karatsuba, also a mathematician, constructed the FEE method.

Work on informatics

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As a student of Lomonosov Moscow State University, Karatsuba attended the seminar of Andrey Kolmogorov and found solutions to two problems set up by Kolmogorov. This was essential for the development of automata theory and started a new branch in Mathematics, the theory of fast algorithms.

Automata

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In the paper of Edward F. Moore,[5]  , an automaton (or a machine)  , is defined as a device with   states,   input symbols and   output symbols. Nine theorems on the structure of   and experiments with   are proved. Later such   machines got the name of Moore machines. At the end of the paper, in the chapter «New problems», Moore formulates the problem of improving the estimates which he obtained in Theorems 8 and 9:

Theorem 8 (Moore). Given an arbitrary   machine  , such that every two states can be distinguished from each other, there exists an experiment of length   that identifies the state of   at the end of this experiment.

In 1957 Karatsuba proved two theorems which completely solved the Moore problem on improving the estimate of the length of experiment in his Theorem 8.

Theorem A (Karatsuba). If   is a   machine such that each two its states can be distinguished from each other then there exists a ramified experiment of length at most  , by means of which one can find the state   at the end of the experiment.
Theorem B (Karatsuba). There exists a   machine, every states of which can be distinguished from each other, such that the length of the shortest experiment finding the state of the machine at the end of the experiment, is equal to  .

These two theorems were proved by Karatsuba in his 4th year as a basis of his 4th year project; the corresponding paper was submitted to the journal "Uspekhi Mat. Nauk" on December 17, 1958 and published in June 1960.[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.

Work on number theory

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The main research works of A. A. Karatsuba were published in more than 160 research papers and monographs.[7][8] [9] [10]

The p-adic method

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A.A.Karatsuba constructed a new  -adic method in the theory of trigonometric sums.[11] The estimates of so-called  -sums of the form

 

led[12] to the new bounds for zeros of the Dirichlet  -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  . A.A. Karatsuba was the first to realize[13] in the  -adic form the «embedding principle» of Euler-Vinogradov and to compute a  -adic analog of Vinogradov  -numbers when estimating the number of solutions of a congruence of the Waring type.

Assume that :   and moreover :   where   is a prime number. Karatsuba proved that in that case for any natural number   there exists a   such that for any   every natural number   can be represented in the form (1) for  , and for   there exist   such that the congruence (1) has no solutions.

This new approach, found by Karatsuba, led to a new  -adic proof of the Vinogradov mean value theorem, which plays the central part in the Vinogradov's method of trigonometric sums.

Another component of the  -adic method of A.A. Karatsuba is the transition from incomplete systems of equations to complete ones at the expense of the local  -adic change of unknowns.[14]

Let   be an arbitrary natural number,  . Determine an integer   by the inequalities  . Consider the system of equations

 
 

Karatsuba proved that the number of solutions   of this system of equations for   satisfies the estimate

 

For incomplete systems of equations, in which the variables run through numbers with small prime divisors, Karatsuba applied multiplicative translation of variables. This led to an essentially new estimate of trigonometric sums and a new mean value theorem for such systems of equations.

The Hua Luogeng problem on the convergency exponent of the singular integral in the Terry problem

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 -adic method of A.A.Karatsuba includes the techniques of estimating the measure of the set of points with small values of functions in terms of the values of their parameters (coefficients etc.) and, conversely, the techniques of estimating those parameters in terms of the measure of this set in the real and  -adic metrics. 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. In 1979 Karatsuba, together with his students G.I. Arkhipov and V.N. Chubarikov obtained a complete solution[15] of the Hua Luogeng problem of finding the exponent of convergency of the integral:

 

where   is a fixed number.

In this case, the exponent of convergency means the value  , such that   converges for   and diverges for  , where   is arbitrarily small. It was shown that the integral   converges for   and diverges for  .

At the same time, the similar problem for the integral was solved:   where   are integers, satisfying the conditions :  

Karatsuba and his students proved that the integral   converges, if   and diverges, if  .

The integrals   and   arise in the studying of the so-called Prouhet–Tarry–Escott problem. Karatsuba and his students obtained a series of new results connected with the multi-dimensional analog of the Tarry problem. In particular, they proved that if   is a polynomial in   variables ( ) of the form :   with the zero free term,  ,   is the  -dimensional vector, consisting of the coefficients of  , then the integral :   converges for  , where   is the highest of the numbers  . 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   (I. A. Ikromov, M. A. Chahkiev and others).

Multiple trigonometric sums

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In 1966–1980, Karatsuba developed[16][17] (with participation of his students G.I. Arkhipov and V.N. Chubarikov) the theory of multiple Hermann Weyl trigonometric sums, that is, the sums of the form

  , where   ,

  is a system of real coefficients  . The central point of that theory, as in the theory of the Vinogradov trigonometric sums, is the following mean value theorem.

Let   be natural numbers,  , . Furthermore, let   be the  -dimensional cube of the form ::   ,  , in the euclidean space : and ::   . : Then for any   and   the value   can be estimated as follows
  , :

where   ,  ,   ,  , and the natural numbers   are such that: ::   ,   .

The mean value theorem and the lemma on the multiplicity of intersection of multi-dimensional parallelepipeds form the basis of the estimate of a multiple trigonometric sum, that was obtained by Karatsuba (two-dimensional case was derived by G.I. Arkhipov[18]). Denoting by   the least common multiple of the numbers   with the condition  , for   the estimate holds

  ,

where   is the number of divisors of the integer  , and   is the number of distinct prime divisors of the number  .

The estimate of the Hardy function in the Waring problem

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Applying his  -adic form of the Hardy-Littlewood-Ramanujan-Vinogradov method to estimating trigonometric sums, in which the summation is taken over numbers with small prime divisors, Karatsuba obtained[19] a new estimate of the well known Hardy function   in the Waring's problem (for  ):

 

Multi-dimensional analog of the Waring problem

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In his subsequent investigation of the Waring problem Karatsuba obtained[20] the following two-dimensional generalization of that problem:

Consider the system of equations

  ,   ,

where   are given positive integers with the same order or growth,  , and   are unknowns, which are also positive integers. This system has solutions, if   , and if  , then there exist such  , that the system has no solutions.

The Artin problem of local representation of zero by a form

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Emil Artin had posed the problem on the  -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. This was shown not to be the case by an example of Guy Terjanian. 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. Karatsuba and his student Arkhipov proved,[21] that for any natural number   there exists  , such that for any   there is a form with integral coefficients   of degree smaller than  , the number of variables of which is  ,  ,

 

which has only trivial representation of zero in the 2-adic numbers. They also obtained a similar result for any odd prime modulus  .

Estimates of short Kloosterman sums

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Karatsuba developed[22][23][24] (1993—1999) a new method of estimating short Kloosterman sums, that is, trigonometric sums of the form

 

where   runs through a set   of numbers, coprime to  , the number of elements   in which is essentially smaller than  , and the symbol   denotes the congruence class, inverse to   modulo  :  .

Up to the early 1990s, the estimates of this type were known, mainly, for sums in which the number of summands was higher than   (H. D. Kloosterman, I. M. Vinogradov, H. Salié, L. Carlitz, S. Uchiyama, A. Weil). The only exception was the special moduli of the form  , where   is a fixed prime and the exponent   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   is an arbitrarily small fixed number. The final paper of Karatsuba on this subject[25] was published posthumously.

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

  • finding asymptotics of the sums of fractional parts of the form :   : where   runs, one after another, through the integers satisfying the condition  , and   runs through the primes that do not divide the module   (Karatsuba);
  • finding a lower bound for the number of solutions of inequalities of the form :   : in the integers  ,  , coprime to  ,   (Karatsuba);
  • the precision of approximation of an arbitrary real number in the segment   by fractional parts of the form :

  : where  ,  ,   (Karatsuba);

  : where   is the number of primes  , not exceeding   and belonging to the arithmetic progression   (J. Friedlander, H. Iwaniec);

  • a lower bound for the greatest prime divisor of the product of numbers of the form :

 ,   (D. R. Heath-Brown);

  • proving that there are infinitely many primes of the form:

  (J. Friedlander, H. Iwaniec);

  • combinatorial properties of the set of numbers :

    (A. A. Glibichuk).

The Riemann zeta function

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The Selberg zeroes

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In 1984 Karatsuba proved,[26][27] that for a fixed   satisfying the condition  , a sufficiently large   and  ,  , the interval   contains at least   real zeros of the Riemann zeta function  .

The special case   was proven by Atle Selberg earlier in 1942.[28] The estimates of Atle Selberg and Karatsuba can not be improved in respect of the order of growth as  .

Distribution of zeros of the Riemann zeta function on the short intervals of the critical line

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Karatsuba also obtained [29] a number of results about the distribution of zeros of   on «short» intervals of the critical line. He proved that an analog of the Selberg conjecture holds for «almost all» intervals  ,  , where   is an arbitrarily small fixed positive number. Karatsuba developed (1992) a new approach to investigating zeros of the Riemann zeta-function on «supershort» intervals of the critical line, that is, on the intervals  , the length   of which grows slower than any, even arbitrarily small degree  . In particular, he proved that for any given numbers  ,   satisfying the conditions   almost all intervals   for   contain at least   zeros of the function  . This estimate is quite close to the one that follows from the Riemann hypothesis.

Zeros of linear combinations of Dirichlet L-series

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Karatsuba developed a new method [30][31] of investigating zeros of functions which can be represented as linear combinations of Dirichlet  -series. The simplest example of a function of that type is the Davenport-Heilbronn function, defined by the equality

 

where   is a non-principal character modulo   ( ,  ,  ,  ,  ,   for any  ),

 

For   Riemann hypothesis is not true, however, the critical line   contains, nevertheless, abnormally many zeros.

Karatsuba proved (1989) that the interval  ,  , contains at least

 

zeros of the function  . Similar results were obtained by Karatsuba also for linear combinations containing arbitrary (finite) number of summands; the degree exponent   is here replaced by a smaller number  , that depends only on the form of the linear combination.

The boundary of zeros of the zeta function and the multi-dimensional problem of Dirichlet divisors

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To Karatsuba belongs a new breakthrough result [32] in the multi-dimensional problem of Dirichlet divisors, which is connected with finding the number   of solutions of the inequality   in the natural numbers   as  . For   there is an asymptotic formula of the form

  ,

where   is a polynomial of degree  , the coefficients of which depend on   and can be found explicitly and   is the remainder term, all known estimates of which (up to 1960) were of the form

  ,

where  ,   are some absolute positive constants.

Karatsuba obtained a more precise estimate of  , in which the value   was of order   and was decreasing much slower than   in the previous estimates. Karatsuba's estimate is uniform in   and  ; in particular, the value   may grow as   grows (as some power of the logarithm of  ). (A similar looking, but weaker result was obtained in 1960 by a German mathematician Richert, whose paper remained unknown to Soviet mathematicians at least until the mid-seventies.)

Proof of the estimate of   is based on a series of claims, essentially equivalent to the theorem on the boundary of zeros of the Riemann zeta function, obtained by the method of Vinogradov, that is, the theorem claiming that   has no zeros in the region

  .

Karatsuba found [33](2000) the backward relation of estimates of the values   with the behaviour of   near the line  . In particular, he proved that if   is an arbitrary non-increasing function satisfying the condition  , such that for all   the estimate

 

holds, then   has no zeros in the region

 

(  are some absolute constants).

Estimates from below of the maximum of the modulus of the zeta function in small regions of the critical domain and on small intervals of the critical line

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Karatsuba introduced and studied [34] the functions   and  , defined by the equalities

 

Here   is a sufficiently large positive number,  ,  ,  ,  . Estimating the values   and   from below shows, how large (in modulus) values   can take on short intervals of the critical line or in small neighborhoods of points lying in the critical strip  . The case   was studied earlier by Ramachandra; the case  , where   is a sufficiently large constant, is trivial.

Karatsuba proved, in particular, that if the values   and   exceed certain sufficiently small constants, then the estimates

 

hold, where   are certain absolute constants.

Behaviour of the argument of the zeta-function on the critical line

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Karatsuba obtained a number of new results[35][36] related to the behaviour of the function  , which is called the argument of Riemann zeta function on the critical line (here   is the increment of an arbitrary continuous branch of   along the broken line joining the points   and  ). Among those results are the mean value theorems for the function   and its first integral   on intervals of the real line, and also the theorem claiming that every interval   for   contains at least

 

points where the function   changes sign. Earlier similar results were obtained by Atle Selberg for the case  .

The Dirichlet characters

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Estimates of short sums of characters in finite fields

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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. Let   be a fixed integer,   a polynomial, irreducible over the field   of rational numbers,   a root of the equation  ,   the corresponding extension of the field  ,   a basis of  ,  ,  ,  . Furthermore, let   be a sufficiently large prime, such that   is irreducible modulo  ,   the Galois field with a basis  ,   a non-principal Dirichlet character of the field  . Finally, let   be some nonnegative integers,   the set of elements   of the Galois field  ,

  ,

such that for any  ,  , the following inequalities hold:

  .

Karatsuba proved that for any fixed  ,  , and arbitrary   satisfying the condition

 

the following estimate holds:

 

where  , and the constant   depends only on   and the basis  .

Estimates of linear sums of characters over shifted prime numbers

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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   on a sequence of shifted prime numbers, namely, an estimate of the form

 

where   is an integer satisfying the condition  ,   an arbitrarily small fixed number,  , and the constant   depends on   only.

This claim is considerably stronger than the estimate of Vinogradov, which is non-trivial for  .

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  , which give a decreased power compared to   compared to  ,  , where   is the modulus of the character. This estimate is of crucial importance, as it is so deep that gives more than the extended Riemann hypothesis, and, it seems, in that directions is a deeper fact than that conjecture (if the conjecture is true). Recently this estimate was improved by A.A.Karatsuba».

This result was extended by Karatsuba to the case when   runs through the primes in an arithmetic progression, the increment of which grows with the modulus  .

Estimates of sums of characters on polynomials with a prime argument

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Karatsuba found [37][39] a number of estimates of sums of Dirichlet characters in polynomials of degree two for the case when the argument of the polynomial runs through a short sequence of subsequent primes. Let, for instance,   be a sufficiently high prime,  , where   and   are integers, satisfying the condition  , and let   denote the Legendre symbol, then for any fixed   with the condition   and   for the sum  ,

 

the following estimate holds:

 

(here   runs through subsequent primes,   is the number of primes not exceeding  , and   is a constant, depending on   only).

A similar estimate was obtained by Karatsuba also for the case when   runs through a sequence of primes in an arithmetic progression, the increment of which may grow together with the modulus  .

Karatsuba conjectured that the non-trivial estimate of the sum   for  , which are "small" compared to  , remains true in the case when   is replaced by an arbitrary polynomial of degree  , which is not a square modulo  . This conjecture is still open.

Lower bounds for sums of characters in polynomials

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Karatsuba constructed [40] an infinite sequence of primes   and a sequence of polynomials   of degree   with integer coefficients, such that   is not a full square modulo  ,

 

and such that

 

In other words, for any   the value   turns out to be a quadratic residues modulo  . This result shows that André Weil's estimate

 

cannot be essentially improved and the right hand side of the latter inequality cannot be replaced by say the value  , where   is an absolute constant.

Sums of characters on additive sequences

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Karatsuba found a new method,[41] making it possible to obtain rather precise estimates of sums of values of non-principal Dirichlet characters on additive sequences, that is, on sequences consisting of numbers of the form  , where the variables   and   runs through some sets   and   independently of each other. 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. Let   be an arbitrarily small fixed number,  ,   a sufficiently large prime,   a non-principal character modulo  . Furthermore, let   and   be arbitrary subsets of the complete system of congruence classes modulo  , satisfying only the conditions  ,  . Then the following estimate holds:

 

Karatsuba's method makes it possible to obtain non-trivial estimates of that sort in certain other cases when the conditions for the sets   and  , formulated above, are replaced by different ones, for example:  ,  

In the case when   and   are the sets of primes in intervals  ,   respectively, where  ,  , an estimate of the form

 

holds, where   is the number of primes, not exceeding  ,  , and   is some absolute constant.

Distribution of power congruence classes and primitive roots in sparse sequences

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Karatsuba obtained[42] (2000) non-trivial estimates of sums of values of Dirichlet characters "with weights", that is, sums of components of the form  , where   is a function of natural argument. Estimates of that sort are applied in solving a wide class of problems of number theory, connected with distribution of power congruence classes, also primitive roots in certain sequences.

Let   be an integer,   a sufficiently large prime,  ,  ,  , where  , and set, finally,

 

(for an asymptotic expression for  , see above, in the section on the multi-dimensional problem of Dirichlet divisors). For the sums   and   of the values  , extended on the values  , for which the numbers   are quadratic residues (respectively, non-residues) modulo  , Karatsuba obtained asymptotic formulas of the form

  .

Similarly, for the sum   of values  , taken over all  , for which   is a primitive root modulo  , one gets an asymptotic expression of the form

  ,

where   are all prime divisors of the number  .

Karatsuba applied his method also to the problems of distribution of power residues (non-residues) in the sequences of shifted primes  , of the integers of the type   and some others.

Late work

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In his later years, apart from his research in number theory (see Karatsuba phenomenon,[43] Karatsuba studied certain problems of theoretical physics, in particular in the area of quantum field theory. Applying his ATS theorem and some other number-theoretic approaches, he obtained new results[44] in the Jaynes–Cummings model in quantum optics.

Awards and titles

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  • 1981: P.L.Tchebyshev Prize of Soviet Academy of Sciences
  • 1999: Distinguished Scientist of Russia
  • 2001: I.M.Vinogradov Prize of Russian Academy of Sciences
 
In Crimea

See also

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References

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  1. ^ "In Memory Anatolii Alekseevich Karatsuba" (PDF). Izvestiya: Mathematics. 72 (6): 1061. 2008. Bibcode:2008IzMat..72.1061.. doi:10.1070/IM2008v072n06ABEH002428. S2CID 250777582.
  2. ^ a b c "Anatolii Alekseevich Karatsuba (On his 60th birthday)". Russian Mathematical Surveys. 53 (2): 419–422. 1998. Bibcode:1998RuMaS..53..419.. doi:10.1070/RM1998v053n02ABEH000013. S2CID 250847741.
  3. ^ D. Knuth, TAOCP vol. II, sec. 4.3.3
  4. ^ List of research works, Anatolii Karatsuba, Steklov Mathematical Institute (accessed March 2012).
  5. ^ Moore, E. F. (1956). "Gedanken-experiments on Sequential Machines". In C E Shannon; J McCarthy (eds.). Automata Studies. Annals of Mathematical Studies. Vol. 34. Princeton, N.J.: Princeton University Press. pp. 129–153.
  6. ^ Karatsuba, A. A. (1960). "Solution of one problem from the theory of finite automata". Usp. Mat. Nauk. 15 (3): 157–159.
  7. ^ Karatsuba, A. A. (1975). Principles of analytic number theory. Moscow: Nauka.
  8. ^ G. I. Archipov, A. A. Karatsuba, V. N. Chubarikov (1987). Theory of multiple trigonometric sums. Moscow: Nauka.{{cite book}}: CS1 maint: multiple names: authors list (link)
  9. ^ A. A. Karatsuba, S. M. Voronin (1994). The Riemann Zeta Function. Moscow: Fiz.Mat.Lit. ISBN 3110131706.
  10. ^ Karatsuba, A. A. (1995). Complex analysis in number theory. London, Tokyo: C.R.C. ISBN 0849328667.{{cite book}}: CS1 maint: location missing publisher (link)
  11. ^ Archipov G.I., Chubarikov V.N. (1997). "On the mathematical works of Professor A.A. Karatsuba". Proc. Steklov Inst. Math. (218): 7–19.
  12. ^ Karatsuba, A. A. (1961). "Estimates of trigonometric sums of a special form and their applications". Dokl. Akad. Nauk SSSR. 137 (3): 513–514.
  13. ^ Karatsuba, A. A. (1962). "The Waring problem for the congruence modulo the number which is equal to the prime in power". Vestn. Mosk. Univ. 1 (4): 28–38.
  14. ^ Karatsuba, A. A. (1965). "On the estimation of the number of solutions of certain equations". Dokl. Akad. Nauk SSSR. 165 (1): 31–32.
  15. ^ G. I. Archipov, A. A. Karatsuba, V. N. Chubarikov (1979). "Trigonometric integrals". Izv. Akad. Nauk SSSR, Ser. Mat. 43 (5): 971–1003.{{cite journal}}: CS1 maint: multiple names: authors list (link)
  16. ^ Karatsuba, A.A. (1966). "The mean value theorems and complete trigonometric sums". Izv. Akad. Nauk SSSR, Ser. Mat. 30 (1): 183–206.
  17. ^ G. I. Archipov, A. A. Karatsuba, V. N. Chubarikov (1987). Theory of multiple trigonometric sums. Moscow: Nauka.{{cite book}}: CS1 maint: multiple names: authors list (link)
  18. ^ Arkhipov, G.I. (1975). "A mean value theorem of the module of a multiple trigonometric sum". Math. Notes. 17 (1): 143–153. doi:10.1007/BF01093850. S2CID 121762464.
  19. ^ Karatsuba, A. A. (1985). "On the function G(n) in Waring's problem". Izv. Akad. Nauk SSSR, Ser. Math. 49 (5): 935–947.
  20. ^ G. I. Archipov, A. A. Karatsuba (1987). "A multidimensional analogue of Waring's problem". Dokl. Akad. Nauk SSSR. 295 (3): 521–523.
  21. ^ G. I. Archipov, A. A. Karatsuba (1981). "On local representation of zero by a form". Izv. Akad. Nauk SSSR, Ser. Mat. 45 (5): 948–961.
  22. ^ Karatsuba, A. A. (1995). "Analogues of Kloostermans sums". Izv. Ross. Akad. Nauk, Ser. Math. 59 (5): 93–102.
  23. ^ Karatsuba, A. A. (1997). "Analogues of incomplete Kloosterman sums and their applications". Tatra Mountains Math. Publ. (11): 89–120.
  24. ^ Karatsuba, A. A. (1999). "Kloosterman double sums". Mat. Zametki. 66 (5): 682–687.
  25. ^ Karatsuba, A. A. (2010). "New estimates of short Kloosterman sums". Mat. Zametki (88:3–4): 347–359.
  26. ^ Karatsuba, A. A. (1984). "On the zeros of the function ζ(s) on short intervals of the critical line". Izv. Akad. Nauk SSSR, Ser. Mat. 48 (3): 569–584.
  27. ^ Karatsuba, A. A. (1985). "On the zeros of the Riemann zeta-function on the critical line". Proc. Steklov Inst. Math. (167): 167–178.
  28. ^ Selberg, A. (1942). "On the zeros of Riemann's zeta-function". SHR. Norske Vid. Akad. Oslo (10): 1–59.
  29. ^ Karatsuba, A. A. (1992). "On the number of zeros of the Riemann zeta-function lying in almost all short intervals of the critical line". Izv. Ross. Akad. Nauk, Ser. Mat. 56 (2): 372–397.
  30. ^ Karatsuba, A. A. (1990). "On the zeros of the Davenport–Heilbronn function lying on the critical line". Izv. Akad. Nauk SSSR, Ser. Mat. 54 (2): 303–315.
  31. ^ Karatsuba, A. A. (1993). "On the zeros of arithmetic Dirichlet series without Euler product". Izv. Ross. Akad. Nauk, Ser. Mat. 57 (5): 3–14.
  32. ^ Karatsuba, A. A. (1972). "Uniform estimate of the remainder in the problem of Dirichlet divisors". Izv. Akad. Nauk SSSR, Ser. Mat. 36 (3): 475–483.
  33. ^ Karatsuba, A. A. (2000). "The multidimensional Dirichlet divisor problem and zero free regions for the Riemann zeta function". Functiones et Approximatio Commentarii Mathematici. 28 (XXVIII): 131–140. doi:10.7169/facm/1538186690.
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