Anatoly Alexeyevich Karatsuba
|Died||28 September 2008 (aged 71)|
|Alma mater||Moscow State University|
Anatoly Alexeyevich Karatsuba (his first name often spelled Anatolii) (Russian: Анато́лий Алексе́евич Карацу́ба; Grozny, Soviet Union, January 31, 1937 – Moscow, Russia, September 28, 2008) 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. there entitled "The method of trigonometric sums and intermediate value theorems" in 1966. He later held a position at the Steklov Institute of Mathematics of the Academy of Sciences.
The main research works of Anatoly Karatsuba were published in more than 160 research papers and monographs.
Awards and Titles
- 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
The early works on Informatics
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.
In the paper of Edward F. Moore, , an automat (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. 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.
Works in Number Theory
The p-adic method
A.A.Karatsuba constructed a new -adic method in the theory of trigonometric sums. The estimates of so-called -sums of the form
led 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 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.
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
-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 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
- , 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). 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
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 a new estimate of the well known Hardy function in the Waring's problem (for ):
Multi-dimensional analog of the Waring problem
In his subsequent investigation of the Waring problem Karatsuba obtained 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
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, 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
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 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);
- a more precise constant in the Brun–Titchmarsh theorem :
- 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:
- combinatorial properties of the set of numbers :
(A. A. Glibichuk).
The Riemann zeta-function
The Selberg Zeroes
Distribution of zeros of the Riemann zeta function on the short intervals of the critical line
Karatsuba also obtained  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
Karatsuba developed a new method  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, anormally 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
To Karatsuba belongs a new breakthrough result  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 (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
Karatsuba introduced and studied  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
Karatsuba obtained a number of new results 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
Estimates of short sums of characters in finite fields
In the end of the sixties Karatsuba, estimating short sums of Dirichlet characters, developed  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
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  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 .
«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
Karatsuba found  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
Karatsuba constructed  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
Karatsuba found a new method, 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
Karatsuba obtained (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.
Works of his later years
In his later years, apart from his research in number theory (see Karatsuba phenomenon, 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 in the Jaynes–Cummings model in quantum optics.
All his life Karatsuba enjoyed many sports: in his younger years, athletics, weightlifting and wrestling, then hiking, rock climbing, caving and mountaineering.
Four times he climbed Mount Elbrus. He hiked in the mountains of Caucasus, Pamir Mountains and, especially in the last years of his life, Tian Shan in Zailiysky Alatau and Teskey Ala-Too. He loved classical music and knew it very well, especially Johann Sebastian Bach and Antonio Vivaldi.
- 1998 Russian Mathematical Survey 53 419 http://iopscience.iop.org/0036-0279/53/2/M21
- D. Knuth, TAOCP vol. II, sec. 4.3.3
- List of research works, Anatolii Karatsuba, Steklov Mathematical Institute (accessed March 2012).
- Moore, E. F. (1956). "Gedanken-experiments on Sequential Machines". In C E Shannon; J McCarthy (eds.). Automata Studies. Annals of Mathematical Studies. 34. Princeton, N.J.: Princeton University Press. pp. 129–153.
- Karatsuba, A. A. (1960). "Solution of one problem from the theory of finite automata". Usp. Mat. Nauk (15:3): 157–159.
- Karatsuba, A. A. (1975). Principles of analytic number theory. Moscow: Nauka.
- G. I. Archipov, A. A. Karatsuba, V. N. Chubarikov (1987). Theory of multiple trigonometric sums. Moscow: Nauka.CS1 maint: multiple names: authors list (link)
- A. A. Karatsuba, S. M. Voronin (1994). The Riemann Zeta Function. Moscow: Fiz.Mat.Lit. ISBN 3110131706.
- Karatsuba, A. A. (1995). Complex analysis in number theory. London, Tokyo: C.R.C. ISBN 0849328667.
- Archipov G.I., Chubarikov V.N. (1997). "On the mathematical works of Professor A.A. Karatsuba". Proceedings Steklov Inst. Math. (218): 7–19.
- Karatsuba, A. A. (1961). "Estimates of trigonometric sums of a special form and their applications". Dokl. Akad. Nauk SSSR (137:3): 513–514.
- 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.
- Karatsuba, A. A. (1965). "On the estimation of the number of solutions of certain equations". Dokl. Akad. Nauk SSSR (165:1): 31–32.
- G. I. Archipov, A. A. Karatsuba, V. N. Chubarikov (1979). "Trigonometric integrals". Izv. Akad. Nauk SSSR, Ser. Mat. (43:5): 971–1003.CS1 maint: multiple names: authors list (link)
- Karatsuba, A.A. (1966). "The mean value theorems and complete trigonometric sums". Izv. Akad. Nauk SSSR, Ser. Mat. (30:1): 183–206.
- G. I. Archipov, A. A. Karatsuba, V. N. Chubarikov (1987). Theory of multiple trigonometric sums. Moscow: Nauka.CS1 maint: multiple names: authors list (link)
- Arkhipov, G.I. (1975). "A mean value theorem of the module of a multiple trigonometric sum". Math. Notes (17:1): 143–153.
- Karatsuba, A. A. (1985). "On the function G(n) in Waring's problem". Izv. Akad. Nauk SSSR, Ser. Math. (49:5): 935–947.
- G. I. Archipov, A. A. Karatsuba (1987). "A multidimensional analogue of Waring's problem". Dokl. Akad. Nauk SSSR (295:3): 521–523.
- 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.
- Karatsuba, A. A. (1995). "Analogues of Kloostermans sums". Izv. Ross. Akad. Nauk, Ser. Math. (59:5): 93–102.
- Karatsuba, A. A. (1997). "Analogues of incomplete Kloosterman sums and their applications". Tatra Mountains Math. Publ. (11): 89–120.
- Karatsuba, A. A. (1999). "Kloosterman double sums". Mat. Zametki (66:5): 682–687.
- Karatsuba, A. A. (2010). "New estimates of short Kloosterman sums". Mat. Zametki (88:3–4): 347–359.
- 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.
- Karatsuba, A. A. (1985). "On the zeros of the Riemann zeta-function on the critical line". Proc. Steklov Inst. Math. (167): 167–178.
- Selberg, A. (1942). "On the zeros of Riemann's zeta-function". SHR. Norske Vid. Akad. Oslo (10): 1–59.
- 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.
- 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.
- Karatsuba, A. A. (1993). "On the zeros of arithmetic Dirichlet series without Euler product". Izv. Ross. Akad. Nauk, Ser. Mat. (57:5): 3–14.
- 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.
- Karatsuba, A. A. (2000). "The multidimensional Dirichlet divisor problem and zero free regions for the Riemann zeta function". Functiones et Approximatio. 28 (XXVIII): 131–140. doi:10.7169/facm/1538186690.
- Karatsuba, A. A. (2004). "Lower bounds for the maximum modulus of the Riemann zeta function on short segments of the critical line". Izv. Ross. Akad. Nauk, Ser. Mat. 68 (68:8): 99–104. Bibcode:2004IzMat..68.1157K. doi:10.1070/IM2004v068n06ABEH000513.
- Karatsuba, A. A. (1996). "Density theorem and the behavior of the argument of the Riemann zeta function". Mat. Zametki (60:3): 448–449.
- Karatsuba, A. A. (1996). "On the function S(t)". Izv. Ross. Akad. Nauk, Ser. Mat. (60:5): 27–56.
- Karatsuba, A. A. (1968). "Character sums and primitive roots in finite fields". Dokl. Akad. Nauk SSSR (180:6): 1287–1289.
- Karatsuba, A. A. (1970). "On estimates of sums of characters". Izv. Akad. Nauk SSSR, Ser. Mat. (34:1): 20–30.
- Karatsuba, A. A. (1975). "Sums of characters in sequences of shifted prime numbers, with applications". Mat. Zametki (17:1): 155–159.
- Karatsuba, A. A. (1973). "Lower estimates of sums of polynomial characters". Mat. Zametki (14:1): 67–72.
- Karatsuba, A. A. (1971). "Distribution of power residues and nonresidues in additive sequences". Dokl. Akad. Nauk SSSR (196:4): 759–760.
- Karatsuba, A. A. (2000). "Weighted character sums". Izv. Ross. Akad. Nauk, Ser. Mat. 64 (64:2): 29–42. Bibcode:2000IzMat..64..249K. doi:10.1070/IM2000v064n02ABEH000283.
- Karatsuba, A. A. (2011). "A property of the set of prime numbers". Russian Mathematical Surveys. 66 (2): 209–220. Bibcode:2011RuMaS..66..209K. doi:10.1070/RM2011v066n02ABEH004739.
- A. A. Karatsuba, E. A. Karatsuba (2009). "A resummation formula for collapse and revival in the Jaynes–Cummings model". J. Phys. A: Math. Theor. 42 (19): 195304, 16. Bibcode:2009JPhA...42s5304K. doi:10.1088/1751-8113/42/19/195304.
- G. I. Archipov; V. N. Chubarikov (1997). "On the mathematical works of professor A. A. Karatsuba". Proc. Steklov Inst. Math. 218.