ACADEMICIAN IVAN MATVEEVICH VINOGRADOV

ACADEMICIAN IVAN MATVEEVICH

VINOGRADOV

To cite this article: A A Karatsuba 1984 Math. USSR Izv. 23 1

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H3B. Ακω. Hayx CCCP Math. USSR Izvestiya Cep. MaTeM. TOM 47(1983), № 4 Vol. 23(1984), No. 1

ACADEMICIAN IVAN MATVEEVICH VINOGRADOV

Soviet science has suffered a severe loss. The prominent Soviet scholar, outstanding scientific organizer, Director of the Steklov Institute of Mathematics, and twice Hero of Socialist Labor, Academician I. M. Vinogradov, passed away at the age of 92 on March 20, 1983.

Associated with Vinogradov's name are great strides in the evolution of the mathematics of this country and the world, the creation and development of new and powerful methods in analytic number theory which have had enormous influence on the development of many areas of mathematics.

Vinogradov was born on September 14, 1891 in the Velikie Luki District of Pskov Province. After finishing in the Department of Physics and Mathematics of Petrograd University in 1914, he was retained at the university to prepare for an academic career. In 1920 he became a professor at the Polytechnic Institute, and then a professor at Petrograd University.

All his research was connected with number theory. A typical feature of Vinogradov's scientific creativity was the treatment of difficult classical problems that had not been solved in the course of centuries. A new method he originated in analytic number theory enabled him to solve many fundamental problems in this theory.

In 1929 Vinogradov was elected a Full Member of the Academy of Sciences of the USSR, and began his intense scientific organizational activity, which he carried on simultaneously with active research. From 1934 to the end of his life he headed the Steklov Institute of Mathematics.

Under his guidance the Institute became a world-famous center of contemporary mathematics. Large centers of Soviet mathematical science were created on the basis of the subdivisions of the Institute.

Vinogradov was the acknowledged head of the Soviet mathematical school. He was the permanent chairman of the National Committee of Soviet Mathematicians. He wrote the celebrated textbook "Foundations of number theory", on which whole generations of mathematicians have been raised.

His scientific merits received broad international recognition. He was chosen as a member of more than 20 foreign academies and scientific societies.

The merits of Academician Vinogradov were highly valued by the Communistic Party and the Soviet government. He was twice honored by the title of Hero of Socialist Labor,

1980 Mathematics Subject Classification. Primary 10A70, 10-03.

©1984 American Mathematical Society 0025-5726/84 $1.00 + $.25 per page

was decorated with five Orders of Lenin, an Order of the October Revolution, and other government prizes, and was a Lenin Prize and State Prize Laureate. The highest prize of the Academy of Sciences of the USSR—the Lomonosov Gold Medal—was awarded to him.

Vinogradov's selfless service to science and the Soviet nation and his devotion to his homeland won him well-deserved esteem and great authority. The fond memory of the outstanding Soviet scholar Ivan Matveevich Viniogradov will forever be preserved in the hearts of the Soviet people.

Yu. V. Andropov, G. A. Aliev, M. S. Gorbachev, V. V. Grishin, A. A. Gromyko, D. A. Kunaev, A. Ya. Pel'she, G. V. Romanov, N. A. Tikhonov, D. F. Ustinov, K. U. Chernenko, V. V. Shcherbitsku, P. N. Demichev, V. I. Dolgikh, V. V. Kuznetsov, Β. Ν. Ponomarev, Sh. R. Rashidov, M. S. Solomentsev, E. A. Shevardnadze,

M. V. Zimyanin, I. V. Kapitonov, K. V. Rusakov, N. I. Ryzhkov,

S. P. Trapeznikov, G. I. Marchuk, A. P. Aleksandrov, V. A. Kotel'nikov, E. P. Velikhov, V. A. Koptyug, A. A. Logunov, Yu. A. Ovchinnikov, P. N. Fedoseev, A. L. Yanshin, S. G. Shcherbakov, G. K. Skryabin, N. N. Bogolyubov, V. S. Vladimirov, L. S. Pontryagin, S. M. Nikol'skn, L. D. Faddeev, A. A. Doroditsyn, A. N. Kolmogorov, S. L. Sobolev, A. N. Tikhonov, E. F. Mishchenko

Ivan Matveevich Vinogradov was born on September 14,1891 in the village of Milolyub in the Velikie Luki District of Pskov Province. His father was the village priest, and his mother was a teacher. Vinogradov received his secondary education in the town of Velikie Luki. He then became a student in the Department of Physics and Mathematics of St. Petersburg University. There he studied in the Mathematics Section and specialized in number theory.

In 1914 Vinogradov finished the university and was retained there to be trained for an academic career, on the basis of his work on the distribution of quadratic residues and nonresidues. On the initiative of V. A. Steklov he was granted a stipend in 1915.

While preparing for very extensive Master's examinations and then successfully passing them, Vinogradov undertook the solution of the most difficult problems in number theory, and already in 1914-1918 did work not less powerful than that of the most prominent number theoreticians of the time. However, far from all of his first papers were oppor-tunely published. Besides, communications between Soviet scientists and their foreign colleagues broke down in the first years after the Revolution, foreign scientific journals were not received by Soviet scientists, and the accomplishments of Soviet mathematicians were not known outside the country, for example, we did not have timely knowledge of the work of Weyl on estimates of trigonometric sums, the work of Hardy and Littlewood on Waring's problem, and other work, while the first papers of Vinogradov were not known abroad. Still, some of them nevertheless came to the West in roundabout ways; so it was, for example, with his paper on the number of lattice points in domains in the plane and in space, published in 1918 by the Khar'kov Mathematical Society. Reprints of the

ACADEMICIAN IVAN MATVEEVICH VINOGRADOV 3 paper were made in 1917 (with that year indicated) and, as it turned out, were then sent abroad by Ya. V. Uspenskii, in particular, to Professor Edmund Landau in Gottingen.

In 1918-1920 Vinogradov worked in Perm, first as a lecturer and then as a professor at Perm State University. At the end of 1920 he returned to Petrograd and became a professor at the Polytechnic Institute, and somewhat later a professor at St. Petersburg State University. At the Polytechnic Institute he offered an originally structured course in higher mathematics, and then at the university a course in number theory which was the basis for his famous textbook "Foundations of number theory". This book (translations of which have been published in many countries) not only acquaints the reader with the elements of number theory in a very short space, actually beginning from the foundations —the definition of the divisibility of one integer by another—but also contains problems (with ways of solving them indicated) that bring the reader in close contact with complicated questions in contemporary science. In 1981 the ninth and most modern edition of this celebrated text was published.

Simultaneously with his activities as a professor, Vinogradov carried out intensive research. In particular, he developed methods enabling him to solve new additive problems in number theory, gave estimates of trigonometric sums more general than those of Weyl, and solved other problems.

His work began to be known abroad. Landau's famous "Vorlesungen iiber Zahlentheorie" (1927) contains a chapter entitled "Die Winogradoffsche Methode", and Vinogradov's results received the highest appraisal from researchers.

In January 1929 Vinogradov was elected a Full Member of the Academy of Sciences of the USSR; thus began his intense scientific organizational activity, which he carried on simultaneously with his scientific research. He and S. I. Vavilov worked out plans for a radical reorganization of the Institute of Physics and Mathematics of the Academy of Sciences of the USSR, and then Vinogradov, in his capacity as Director, assumed the leadership of the Mathematics Branch of the Institute. In 1934 the Institute of Physics and Mathematics was divided into two independent institutes: the Steklov Institute of Mathematics and the Lebedev Institute of Physics. Academician Vinogradov was con-firmed as Director of the Institute of Mathematics and held the post continuously until the end of his life. He was a member of the editorial board of the journal "Izvestiya Akademii Nauk SSSR, Otdelenie Matematischeskikh i Estestvennykh Nauk, Seriya Matematicheskaya" from 1937 to 1948; in 1948 he became Editor, and in 1950 Editor-in-Chief of the journal "Izvestiya Akademii Nauk SSSR, Seriya Matematicheskaya".

Vinogradov's first scientific article was written in 1914 under the guidance of Uspenskii, and was on the theory of quadratic residues. The foundations of the theory were laid by Euler, Gauss, and Legendre and make up the main part of the contemporary elementary number theory. The problem is as follows. A prime number ρ > 2 (called a modulus) is given, and all the integers not divisible by ρ are broken up into two classes: the first class contains the integers which give the same remainders when divided by ρ as do squares of integers, and the second class contains the remaining numbers. The numbers of the first class are called quadratic residues modulo p, and those of the second class are called quadratic nonresidues modulo/». For example, if ρ = 11, then the numbers 1, 3, 4, 5, and 9 are quadratic residues modulo 11, while the numbers 2, 6, 7, 8, and 10 are quadratic nonresidues; all other quadratic residues (respectively, nonresidues) are obtained by adding integer multiples of 11 to the given ones. Since the remainders after division by ρ

repeat periodically, in studying quadratic residues and nonresidues we can confine ourselves to the smallest ones, i.e., to the numbers 1,2,... ,p - 1, and ask the appropriate questions about the quadratic residues and nonresidues among just these numbers.

The first theorem here was the Euler-Gauss theorem, which asserts that exactly (p - l)/2 of the numbers 1,2,...,ρ - 1 are quadratic residues and (p - l)/2 are quadratic nonresidues. A useful concept in these questions is the concept of the " Legendre symbol" (introduced by Legendre), denoted by (n/p) and defined for all integers η as follows:

!

+1 if η is a quadratic residue modulo ρ, 0 if η is a multiple of ρ,

- 1 , if η is a quadratic nonresidue modulo ρ.

Thus, the Euler-Gauss theorem implies that the sum of the Legendre symbols over all η from some a to a + ρ — 1 is zero. An important theorem in the theory as a whole is "Gauss's law of reciprocity": if ρ > 2 and q > 2 are prime numbers, then

q ι\ρ

Gauss gave several proofs of this assertion. One of them is based on his formula

n\ 1 £ Ik' _{•ikn/p^ (1)}

\y ι ι\μ) k = 1\ y/

where

In this formula i1 = — 1, and for a real number φ

e"p = cos φ = i'sin<p.

Therefore, the sum on the right-hand side of (1) and generalizations of it are now called " trigonometric sums".

Vinogradov was asked to give a similar proof of Gauss's law of reciprocity if possible. It should be noted that in his university training he studied probability theory under A. A. Markov with great interest, and he "knew Markov's course by heart". Therefore, he perceived quadratic residues and nonresidues as elementary events, and from this arose the following problem: let η = Ν + 1, Ν + 2,... ,Ν + Μ (i.e., M trials are carried out); among these numbers, how many are quadratic residues and how many are nonresidues? In other words, how many times is the Legendre symbol (n/p) equal to + 1 and how many times is it equal to -1? Of course, as happens in probability theory, any conformity with a law in this question emerges for a large number of trials, i.e., ρ and Μ should be regarded as sufficiently large numbers.

Letting V(M) denote the number of quadratic residues of the form 0 < iV + 1, Ν + 2,..., Ν + Μ < ρ, one arrives at the equality

ACADEMICIAN IVAN MATVEEVICH VINOGRADOV where

N + M

/* τ ινί ,

S(M) =

Σ

The problem was thereby reduced to the study of the sum S(M), and, in the first place, to an upper estimate of its modulus. In his 1914 paper Vinogradov proved the following theorem:

For any Μ and Ν

\S{M)\<Jp~\np. (3)

From this and from (2) it follows that the numbers of quadratic residues and nonresidues modulo ρ are asymptotically equal on any interval of length Μ if Μ > \Jp~ lnc ρ for some

e > 1.

The estimate (3) was also obtained by Polya in 1918. This estimate and generalizations of it to sums of "characters" are now called " Vinogradov's estimates of a character sum", and have been used up to the present time in diverse investigations in number theory.

The famous "Vinogradov conjecture on the least nonresidue" modulo ρ appeared at roughly the same time: for any ε > 0 there exists a p0 = po(e) such that for any ρ > p0 the

least nonresidue modulo ρ does not exceedρε. This conjecture has not yet been proved.

The estimate (3) follows immediately from (1). Despite the simplicity of the derivation of (3), the estimate has not been essentially improved. The fact is that there are infinitely many prime numbers/? and numbers Μ such that

\S(M)\> Jplnlnp.

Moreover, if ο(·/ρ In p) is obtained on the right-hand side of (3), then the Vinogradov conjecture on the least nonresidue world follow at once.

Somewhat later (1926) Vinogradov came back to the theme of the distribution of nonresidues and proved the following theorem by using (2) and again a simple but clever argument (if η is the least nonresidue modulo/?, then only numbers divisible by primes not less than η can be the nonresidues up to ^fp In2 p): for ρ > p0

n<p1/cln2p, c^lie.

Vinogradov returned several times to the study of quadratic residues and nonresidues and to generalizations of the problem. This was a favorite topic of Linnik, who was not able to obtain any new results right here. However, he modified the problem in such a way (he started considering the "exceptional moduli p, i.e., those ρ such that the Vinogradov conjecture did not hold) that he could solve it completely in the new formulation (the Vinogradov conjecture and even a sharper assertion are valid for almost all p). The method of solving the latter problem was later extensively developed and applied to the solution of other problems in number theory, and it came to be known as the "large sieve" method.

The next cycle of Vinogradov's work relates to the asymptotic behavior of the number of "lattice points" in planar domains. In its simplest form the problem is as follows.

A lattice point in the plane with a Cartesian system of coordinates X°Y is defined to be a point Μ whose coordinates x, y are integers; let T(R) be the number of lattice points lying in

the disk x2 + y2 < R. Then

T(R) = vR + A(R)

as R -> +00. Find the best possible upper bound for

Gauss posed the problem of the number of lattice points in the disk and obtained the first result here, which can be stated as

The problem of the number of lattice points with positive coordinates under the hyperbola xy = R (the "divisor problem" of Dirichlet) can be posed similarly (and solved similarly).

In 1903 Voronoi worked out a method he used to prove that the remainder term in Dirichlet's asymptotic formula expressing the number of lattice points with positive coordinates under a hyperbola does not exceed the cube root of the principal term in order (the Voronoi method with the analogous result was carried over to the Gauss problem by Sierpihski).

Vinogradov (in 1917) considered the more general problem of finding asymptotic formulas for the number of lattice points in arbitrary planar domains. He developed a new arithmetic method he could use to prove a theorem on the number of lattice points in planar domains that can be made up of finitely many curvilinear trapezoids of the form χ = α, χ = b, y = 0, y = f{x), where it is required only that f"(x) be continuous and have a certain order of growth for a < χ =ξ b (this includes, in particular, disk and hyperbola domains). The corresponding theorem on the number of lattice points in planar domains bounded by curvilinear trapezoids goes as follows:

Suppose that the function y = f(x) is twice continuously differentiable, and that 1/A<&f"(x)<l/A, A » 1,

on the interval [a, b]. Then the number of lattice points in the planar domain bounded by the lines χ = α, χ = b andy — 0 and the curve y = f(x) differs from the area of this domain by a quantity not exceeding R, where

Jarnik later proved that to within a logarithmic factor the remainder term in this formula is best possible on the class of domains being considered.

Here Vinogradov worked out in a rough but perfectly clear form a method for replacing the trigonometric sum

S= Σ

e

"

,

by a 'shorter" sum (the main idea of this substitution was contained in the cited work of Voronoi; a special case of the same formula was considered by Hardy and Littlewood in 1914 in their work on the derivation of an approximate functional equation for the Riemann zeta function).

There is a form of the "substitution formula" that is convenient for applications and more modern in Vinogradov's monograph "Basic variants of the method of trigonometric sums" (1976):

ACADEMICIAN IVAN MATVEEVICH VINOGRADOV 7 Let H, U,A, q, and r be real numbers such that

Η > 0, £ / » Λ » 1 , 0 < r < q < U.

Suppose, further, that f'(x) and <p(x) are real algebraic functions of bounded degree, and that in the interval q < χ < r

1 .„. ν 1 , , „ , , 1 φ(χ) < Η, φ'(χ) « ~j , φ"(χ) « -^. Then

Σ

where, with xn determined by the equality f'{xn) = n,

^n /χ- ι ..., r- '

and bn = 1 if η is different from f'{q) and f'(r) and bn = 1/2 otherwise. It is always true

that Zn *§; H\[A . Finally, Tq equals zero iff'(q) is an integer and equals

otherwise, while Tr equals zero iff'(r) is an integer and equals

otherwise.

Vinogradov later returned several times to problems involving lattice points. The following is the most significant of the theorems proved here (1963):

The number Τ of lattice points in the ball domain x2 + y2 + z2 < R2 can be expressed by

the formula

T=§irR3 + O(R^3 In6 R).

In proving this complicated theorem he repeatedly applied his "substitution formula", and at the last stage of the arguments estimated a trigonometric sum by using only the simplest variant of this formula, i.e., by using only the order of growth of the second derivative of the function in the exponent. Of course, if one uses the "substitution formula" also at the stage and estimates the "short" sum which then appears by using the order of growth of the &th derivative of the function in the exponent (& is some fixed number chosen in an optimal way), then the exponent 4/3 in the remainder term is replaced by a smaller number.

In 1924 Vinogradov began to work on additive problems in number theory. The first principal result here was a new solution of Waring's problem, which was found in 1927 and served as a basis for the creation of a powerful method in modern number theory: the method of trigonometric sums. In 1770 Waring made a conjecture generalizing the Lagrange theorem on the representability of a natural number as the sum of four squares of integers. This conjecture can be formulated as follows: for any fixed integer η > 3, each

natural number can be represented as the sum of a fixed number of nth powers of natural numbers. The general solution of Waring's problem was given by Hilbert in 1909.

In 1922 Hardy and Littlewood worked out a new method (the so-called "Hardy-Littlewood-Ramanujan circle method") for solving a broad class of additive problems, and with its help gave, in particular, a new proof of the Hilbert-Waring theorem. The Hardy-Littlewood-Ramanujan circle method had as its source Euler's method of generat-ing functions (Euler solved linear additive problems by this method) and was essentially based on the theory of functions of a complex variable (the "generating functions" were taken to be infinite series in the domain of a complex variable, and the Cauchy integral formula was used to compute the coefficients of these series). In 1927 Vinogradov applied finite trigonometric sums to the solution of Waring's problem, and this not only led to a new and quite simple proof, but also opened the way to the solution of other difficult problems in number theory (for example, this method can be used to solve problems on the representation of natural numbers as sums of values of a positive integer-valued polynomial).

The integers are distributed periodically among the real numbers. The simplest trigono-metric functions—the sine and the cosine—are periodic with period 2π and thereby provide the possibility of analytically separating out the integers from the set of all real numbers. Trigonometric sums are finite sums of sines and cosines whose arguments are real integral functions. Formulas expressing the number of solutions of an arbitrary equation in the integers can be written simply enough with the help of integrals of such sums. For example, from the easily derivable equality

, \ 0 if « is a nonzero integer,

it follows that the number J of representations of Ν as a sum x" + • • • + xnk, where

x1,...,xlc are natural numbers, is equal to the integral

J=C Sk(a)e-2"aNda, S(a) = £ e2"iax".

The sums S(a) (and also generalizations of them) are called trigonometric sums.

It is just as simple to formulate problems on the distribution of fractional parts of integral functions in the language of trigonometric sums; to do this it suffices to expand the characteristic function of an interval in a Fourier series. Problems of lattice points in planar and spatial domains can also be reduced to trigonometric sums; to do this it suffices to expand the function "fractional part of x" in a Fourier series. A broad circle of diverse problems in number theory can thereby be formulated uniformly in the language of trigonometric sums.

The general scheme for investigating these number theory problems by Vinogradov's method of trigonometric sums is as follows: one writes out an exact formula expressing the number of solutions of the equation under study, or the number of fractional parts of the function under study that fall in a given interval, or the number of lattice points in a given domain as an integral of a trigonometric sum or as a series whose coefficients are trigonometric sums; the exact formula is represented as the sum of two terms, a principal term and a supplementary term (for example, if the Fourier series of the characteristic function of an interval is being considered, then the principal term is obtained from the zero coefficient of the series); the principal term furnishes the principal term of the

ACADEMICIAN IVAN MATVEEVICH VINOGRADOV 9 asymptotic formula, and the supplementary term is the remainder term. In additive problems like Waring's problem, Goldbach's problem, etc., the principal term can be investigated by a method close to the Hardy-Littlewood-Ramanujan circle method (this method is now called "the Hardy-Littlewood-Ramanujan circle method in the form of Vinogradov trigonometric sums"). In most other problems (the distribution of fractional parts, lattice points in domains, etc.) the principal term can be obtained trivially. The problem now arises of estimating the remainder term, and if it can be proved to be of order less than that of the principal term, then the asymptotic formula will be proved and the problem posed will be solved. The main difficulty in estimating the remainder term is obtaining the sharpest possible estimates of the trigonometric sums appearing here.

In proving Waring's theorem (1927) Vinogradov estimated the trigonometric sums S(a) by the method of Weyl (he called these sums, and even more general ones with an arbitrary polynomial in the exponent, "Weyl sums", and the name has been commonly adopted).

In addition to the Waring problem Vinogradov solved by his method of trigonometric sums the general problem of an asymptotic formula for the number of representations of a natural number as a polynomial

axx" + · · · + arx" + ψ(χ1,...,χΓ),

where φ(Χχ, •. .,xr) is a polynomial of degree at most η — 1.

Continuing to occupy himself with Waring's problem, Vinogradov in 1934 created a new method for estimating trigonometric sums that was sharper by far than Weyl's method. This new method gave him fundamentally more powerful results in the problem of the distribution of the fractional parts of polynomials, in Waring's problem itself, in the problem of approximating a real number by fractional parts of an integral polynomial, and so on. At the same time vinogradov's method was successfully used in the theory of the Riemann zeta function (Chudakov), in the Hilbert-Kamke problem (Mardzhanishvili), and in diverse mixed additive problems.

First of all we dwell on results obtained by Vinogradov in Waring's problem. In investigating this problem Hardy and Littlewood introduced the function G(n): the minimal number of terms sufficient for representing all large natural numbers as a sum of k = G(n) «th powers. They obtained an upper bound of the order n2" for G(n). They proved an asymptotic formula for the number of representations of a natural number as a sum of nth powers, with the number of summands also of the order nl".

In 1934 Vinogradov used his method to obtain an upper estimate of the order η In η for G(n), and this cannot be essentially improved any more, due to the obvious inequality G(n) > n. A little later he proved that the Hardy-Littlewood asymptotic formula in Waring's problem is valid with the number of terms of the order «2log n. In addition, he

obtained fundamentally more powerful results at this time in the problem of estimating Weyl sums and, as a consequence of these estimates, in the problem of the distribution of the fractional parts of a polynomial, in the problem of approximating a given fraction by the fractional part of an integral polynomial, and so on. For example, in estimating trigonometric sums of the form (" Weyl sums")

ρ

he obtained the following result: If for some r with 2 < r < η

and, moreover, Ρ < q =ζ Pr l, then S has the estimate

I j-tl πΛ — η *•

Previously known was an estimate by Weyl's method with a decrease weaker by far, namely, with ρ = 2~B + 1.

The conceptual part of Vinogradov's method was based on the possibility of obtaining nontrivial estimates of double sums of the form

w=

Σ

Σ t(x)t{y)e

*

,

where ξ(χ) and i-(y) are real functions of an arbitrary nature, and α is a real number with

q qi'

(the method of estimating sums of the form W is now known as the "Vinogradov smoothing method").

Moreover, Vinogradov made essential use in his method of a property observed already by Euler of nth powers of natural numbers: the difference of two nth powers of natural numbers is either equal to zero or "large" in absolute value.

The basis for his method of estimating Weyl sums is the "Vinogradov mean value theorem". Trigonometric sums S = S(an,.. . ,α^), as functions of the arguments an,.. .,av

are periodic in each argument, with the same period 1. The integral Jh (now called the

"Vinogradov integral") given by

• I | S ( an, . . . ,0^)1 dan • • • da,

ο -Ό

is the mean value of the 26th power of the modulus of S. The mean value theorem can be formulated as follows:

Let η be an integer, η > 2, ν = 1/n, /(χ) = anx" + · · · + α,χ, Ρ > 1, / a positive

integer, b > nl, and lb dan • • • da,. Then where D, = (ηΐγηΙ(2ηγη(η+1)1/ζ, Δ(/) = η{" + 1) (ΐ - ( 1 - ρ)1). When b > b0, where b0 = [n2(21n« + lnlnn + 2.6)],

ACADEMICIAN IVAN MATVEEVICH VINOGRADOV 11

this theorem gives us a sharp estimate of Jh:

Jh « p2*-(«2 + «)/2)

and this plays the main role in estimating Weyl sums, in the Hilbert-Kamke problem, in deriving the asymptotic formula in Waring's problem, and in many other problems in number theory. The mean value theorem itself enables us to estimate Weyl sums even when the degree η of the polynomial can increase as the basic parameter Ρ increases. Such estimates are used in the problem of the distribution of prime numbers, in the multidimen-sional Dirichlet divisor problem, and so on.

The following statement is an immediate corollary of the mean value theorem:

Let k > 8 be a constant. From the η-dimensional domain Un of points (an,... ,ax), where

0 < αη< 1 , . . . , 0 < α1< 1 , with volume 1 it is possible to select a small domain Ω with volume V satisfying

(c(n, k) is a constant depending only on η and k) such that the points of the remaining part of Tln satisfy

Ρ ο2·ηί{αηχ"Λ +βι

It is not hard to prove that it is not possible to replace the indicated inequality for V by some other inequality

V < c'(n,k)P~g'Mn+1)/2

of the same form but with g' = g(n, k) > 1.

Thus, a very sharp upper estimate holds for all but insignificantly few Weyl sums. Despite the metric character of this statement, it penetrates so deeply into the theory of Weyl sums that it enables us now to prove theorems on estimates of individual trigono-metric sums fairly easily.

Vinogradov's general theorem on estimating a Weyl sum can be stated as follows: Let η > 3 be a constant, and let

Tm = Tm{an,...,ai) = Σ e2*im(°"x"+-+*x).

Let the points (an,...,al) in η-dimensional space be divided into two classes. The points in

the first class have the form

where the first terms are irreducible rational fractions with positive denominators having least common multiple Q at most P1/n, and the second terms satisfy the condition

The second class includes all points not in the first class. If

1 Ρ = _8«

andO < m < P2p, then for points of the second class

\Tm\«P1^.

And if

andO < m < P4/n , then for points of the first class

or

if δ0

The mean value theorem and the estimate of a Weyl sum imply an asymptotic formula for the Vinogradov integral/,,. The corresponding theorem can be formulated as follows:

If b > b0 = 2[/?2(21n η + In In η + 2.6)], then/,, satisfies the asymptotic formula

Jh = θ0σΡ2Ι>-("2+η)/2 + o(P2h-("2+n)/2~s), where 00 = / dx 2b oo

°-

Σ

Σ

(in this formulation the theorem was proved by Hua Lo-keng and bears the name "Tarry's problem").

Van der Corput, Chudakov, Hua, Linnik, Mardzhanishvili and Karatsuba, are among the mathematicians who have taken part in the development of Vinogradov's method and applications of it.

In 1937 Vinogradov created a method for estimating trigonometric sums with prime numbers, i.e., trigonometric sums with summation over primes.

He first found a nontrivial estimate for a "linear" sum with prime numbers:

LetH = e^/2, r = lnN, and

Then

Using this estimate and the method described above for solving additive problems, he proved an asymptotic formula for the number of representations of an odd number as a sum of three primes, and this implied that any sufficiently large odd number is a sum of three prime numbers. He thereby solved the ternary Goldbach problem, which had

ACADEMICIAN IVAN MATVEEVICH VINOGRADOV 13 remained unsolved for two centuries since it first arose in 1742 in Goldbach's correspon-dence with Euler. Vinogradov's celebrated "three primes theorem" can be stated as follows:

The number I(N) of representations of an odd number Ν as a sum of three prime numbers satisfies the asymptotic formula

21n3iV \ In ·5 eN 1

-nii + T^-jlrrii- * •>

where the first product is over all primes and the second is only over the prime divisors of N. Somewhat later Vinogradov estimated more general sums with primes, namely, sums T'm, similar to the Weyl sums

7" = V1 . 2 t i « i ( o , f " + •··+«,/>)

His method for estimating trigonometric sums with prime numbers made it possible to solve several new problems for which there had previously been no methods of approach (for example, Waring's problem for prime numbers, and the Hilbert-Kamke problem for prime numbers).

The following lemma ("Vinogradov's sieve") is the basis of Vinogradov's method for estimating sums with primes along with the method for smoothing double sums mentioned above:

Suppose that 0 < c < 1/6, 0 < σ < 1/3, and Ρ is a product of prime numbers that do not exceed x". If

Z) = ( l n x )l n l n V l n ( 1 + c ),

then all the divisors d of Ρ not exceeding χ can be distributed among fewer than D collections with the following properties:

a) the numbers d in a particular collection have the same number β of prime factors and, consequently, the same value μ(ί/) = ( - 1 ) ^ ;

b) one of the collections {called the elementary collection) consists of the single number d = 1, and for it φ is set equal to I, so that d = φ = 1, while any other collection has its own value of ψ such that all the numbers d in this collection satisfy the condition

ψ < d < <p1 + c;

c) for every collection except the elementary one and for any U with 0 < U =ξ φ there exist two collections {the second of which may be the elementary one) of numbers d' and d" with corresponding values φ' and φ" satisfying

U < φ' < Ux", φ'φ" = ψ

such that for some characteristic positive integer Β the products d'd" with {d\ d") = 1 yield all the numbers d of the chosen collection, each repeated Β times.

In succeeding years Vinogradov repeatedly improved and perfected his method of trigonometric sums. He used the method of estimating trigonometric sums with prime numbers in 1953 to estimate the sum of a nonprincipal character over a sequence of translated prime numbers; the estimate obtained cannot be derived from the most

powerful conjectures on the distribution of primes (for example, from the generalized Riemann hypothesis). This theorem can be stated as follows:

Suppose that Q is a prime number, χ is a nonprincipal character modulo Q, and (k, Q) = 1. Then

Σ X(P + k) « N^'iQ^N-^3 + A

This estimate implies, for example, the following arithmetical result: the numbers of quadratic residues and nonresidues modulo Q of the form ρ + k, ρ ^ x, are asymptotically equal, i.e., are equal to

provided that χ > Q3 / 4 + e. A similar assertion, except for χ > Q1+e, can be obtained from

the generalized Riemann hypothesis on the zeros of the Dirichlet L-function.

In 1957 Vinogradov employed his method for estimating Weyl sums to obtain a new bound for the zeros of the Riemann zeta function and, as a corollary, a new remainder term in the asymptotic formula for the distribution of the primes not exceeding a given bound:

f ^ λ(χ) =

(InIn x)~

·

,

c >

0.

He obtained in 1963 a fundamentally new remainder term in the asymptotic formula for the number of lattice points in a ball.

In the period 1958-1971 Vinogradov proved new general theorems on estimates of Weyl sums, and these enabled him to judge the size of their moduli for any possible values of the coefficients of the polynomial in the exponent (these results are presented mainly in his 1971 monograph "The method of trigonometric sums in number theory", and in the new 1980 edition of it).

Vinogradov's monograph "Special variants of the method of trigonometric sums" appeared in 1976 and contained a systematic presentation of those of his main investiga-tions not included in the previous monograph. There he gave a picture of the origin of his method of trigonometric sums: the solutions of particular problems in number theory gave rise to new ideas and new problems, which, in turn, let to the solution of difficult and general problems and to the creation of a powerful method in analytic number theory. Moreover, this monograph has a detailed account of his research on the problem of lattice points in planar and spatial domains. The theorem is proved that the number Τ of lattice points in the domain x2 + y1 + z2 < a1 can be expressed by the formula

a theorem giving an estimate of G(n) is proved, namely,

G(n) < n(2lnn + 41nln« + 21nlnln« + 13);

theorems are proved on estimates of elementary sums with prime numbers, in particular, the sums

ACADEMICIAN IVAN MATVEEVICH VINOGRADOV 15

and applications of these estimates to arithmetic problems are given; and an elementary method of trigonometric sums with prime numbers is presented.

The methods Vinogradov created for solving problems in number theory have found important applications in diverse areas of mathematics: mathematical analysis and the calculus of approximations, probability theory, and mathematical physics.