VINOGRADOV, IVAN MATVEEVICH

(b. Milolyub, Velikie Luki, Russia, 14 September 1891;

d. Moscow, USSR, 20 March 1983), analytic number theory, science administration.

Vinogradov was one of the most prominent mathematicians who worked in analytical number theory. His methods (first of all the method of trigonometric sums) permitted him to make progress with some of the most important and difficult problems in the subject including Goldbach’s conjecture and the estimate of Weyl sums.

Education and Early Results. Vinogradov’s father, Matvei Avraamievich, was a priest in Milolyub, a village in the Velikie Luki district (now of the Pskov province), and his mother, Mariya Aleksandrovna, was a teacher. Ivan was educated in a high school in Velikie Luki, and in 1910 he entered the Faculty of Mathematics and Physics at St. Petersburg University. Professors Andrei A. Markov and Yakov V. Uspenskii exercised a powerful influence on him, and he became interested in number theory. He graduated in 1914 with the degree candidate of mathematical sciences, and for his outstanding work on the distribution of quadratic residues and nonresidues, supervised by Uspenskii, he was awarded a scholarship in 1915. The scientific degrees were abolished after the revolution of 1917 by the Soviet government and were restored only in 1933. After that, Vinogradov was awarded the degree doctor of mathematical and physical sciences. Between 1914 and 1918 he obtained important results in number theory on the distribution of power nonresidues, he elaborated an elementary method that permitted him to find a general theorem on the number of integral points in plain domains, and he proposed a general analytical method for a solution of the problem of the number of integral points in the domains in the plane and in space, and so forth.

But these were the years of World War I and the Russian Revolution. Many of his results were published very late, and others that were published at the time remained unknown in the West because of wartime difficulties. For the same reason, little information reached Russia about results obtained in the West in those years. For example Hermann Weyl’s work on estimates of trigonometric sums and Godfrey Hardy and John Littlewood’s results on the Waring problem did not became known in good time.

Disorganization and hunger in postrevolutionary Petrograd (the new name for St. Petersburg) forced the scientists to flee to safety. Some of the young mathematicians (including Abram S. Besicovitch, Yakov D. Tamarkin, and Aleksandr A. Friedman) went to Perm: there in 1916 a branch of Petrograd University was opened that in 1917 became an independent university. I. M. Vinogradov was among them. From 1918 to 1920 he worked in the Perm State University (from 1920 as professor) and at the end of 1920 he returned to Petrograd, where he became a professor at Petrograd University (from 1925 he held the chair of number theory) and at the Polytechnic Institute. At the Polytechnic Institute he elaborated an original course of mathematics for engineers (described in his handbook Elementy vysshei matematiki[1932, 1933; Elements of higher mathematics]).

In the university he lectured on number theory, on the basis of which he wrote a famous textbook Osnovy teorii chisel (1936; Elements of number theory). Rather concise, this book introduced the elements of theory, beginning with divisibility theory and proceeding via problems with hints for their solutions to problems in the modern theory. This textbook was revised many times— the last (tenth) edition appeared in 2004—and was translated into many languages.

Trigonometric Sums. Vinogradov combined teaching with very intensive research activity. In 1924 he began to study additive problems in number theory. In 1927 he published a new solution of Waring’s problem. This was the beginning of his famous method of trigonometric sums. At first he utilized Weyl’s method, but already in 1934 he proposed a new and more accurate method for estimating trigonometric sums. With the aid of this method he could significantly improve the available results for the problem of the distribution of fractional parts of the values of a polynomial, for Waring’s problem, and for others. Vinogradov’s method was elaborated and applied with success to many different problems in number theory (the theory of the Riemann zeta function, Hilbert-Kamke’s problem, etc.) by a series of scientists: I. Johannes G. Van der Corput, Nikolai G. Chudakov, Hua Loo-Keng, Yurii V. Linnik, Anatolii A. Karatsuba, and others.

In 1937 Vinogradov developed the method of estimates of trigonometric sums over primes, that is to say of trigonometric sums in which the summation is taken over prime numbers. With this method he solved a series of problems in additive prime number theory. In the same year he demonstrated the asymptotic formula for the number of representations of an odd number as the sum of three primes (Godfrey H. Hardy and John E. Little-wood’s 1923 demonstration of this result was based on a certain hypothesis in the theory of L-series). One of the consequences of this formula became a solution of Gold-bach’s problem for odd numbers: every sufficiently large odd integer is representable as a sum of three primes.

In the following years Vinogradov improved his method of trigonometric sums and consequently he found new results for a series of the problems in number theory: an improved estimate for the sum of nontrivial characters in the sequence of shifted primes (1953), a new bound for zeroes of the Riemann zeta function (1957), a new remainder term in the asymptotic expression for the number of integral points in a sphere (1963), and new general theorems on the estimates of Weyl sums (1958–1971). Many of these results were included in his classic monograph Metod trigonometricheskikh summ v teorii chisel (1947; The Method of Trigonometrical Sums in the Theory of Numbers) and also in his book Osobye varianty metoda trigonometricheskikh summ (Special variants of the method of trigonometric sums). His methods were developed widely by others and found application in different parts of mathematics: in mathematical analysis, in the calculus of approximations, in the theory of probability, and in the discrete mathematics.

Vinogradov’s results quickly brought him universal acknowledgment. A section titled “Vinogradov’s Method” appeared in Edmund Landau’s famous Vorlesungen über Zahlentheorie (Lectures on number theory) published in 1927. In 1929 Vinogradov became an effective member of the Academy of Sciences of the USSR. Later he was elected to several foreign academies of sciences and scientific societies, and received honorary doctorates from many universities.

Leadership. Very early in his career he became involved with scientific administration. In 1932 he became the head of the Mathematical Department of the Physics and Mathematics Institute of the USSR Academy of Sciences, from which Steklov Mathematical Institute was reorganized as separate institution in 1934. Vinogradov became its director. In the same year the presidium of the Academy of Sciences moved from Leningrad (formerly Petro-grad) to Moscow. Together with the presidium several institutes of the academy (including the Steklov Institute) moved to Moscow.

As a result of these moves, the two main Russian mathematical schools—the St. Petersburg–Leningrad and the Moscow schools, which had hitherto been in confrontation—were forced to live together. On the basis of a synthesis of these schools the core of the new organism was born: the Soviet mathematical school, one of the most influential in the twentieth century. The edifice of Soviet science, constructed in the 1930s according to Joseph Stalin’s plans, was crowned by its general staff, that is, by the Academy of Sciences of the USSR. Consequently the Steklov Mathematical Institute held the paramount position in the Soviet mathematical community, and the director of this institute became one of the most influential persons in this community. For more than forty-five years this post was occupied (until his death) by Vinogradov (except 1941–1943, when Sergei L. Sobolev was its director).

Under Vinogradov’s leadership the Steklov Mathematical Institute—which was comparatively small (by Soviet standards) and whose members were, with rare exceptions, prominent specialists in the main directions of the contemporary mathematics—became one of the most important mathematical institutes of the twentieth century. However, it must be said that Vinogradov’s tough line in the selection of the personnel was characterized (especially during the last years of his life) by a remarkable willfulness.

For many years Vinogradov was the president of the bureau of the National Committee of the Soviet Mathematicians and the editor of the mathematical series of the Izvestiya Akademii nauk SSSR (Proceedings of the Academy of Sciences of the USSR). Despite the fact that he was not a member of the Communist Party, the Soviet state always supported him. He received many Soviet honors such as: Hero of the Socialist Labor, on two occasions; Order of Lenin, on five occasions; and the Order of the October Revolution. For his mathematical achievements he received the Stalin Prize, the Lenin Prize, and the Lomonosov Gold Medal of the USSR Academy of Sciences.

BIBLIOGRAPHY

WORKS BY VINOGRADOV

Elementy vysshei matematiki[Elements of higher mathematics]. Vol. 1, Analiticheskaya geometriya[Analytical geometry]. Leningrad: Kubuch, 1932.

Elementy vysshei matematiki[Elements of higher mathematics]. Vol. 2, Differentsial’noe ischislenie[Differential calculus]. Leningrad: Kubuch, 1933.

Osnovy teorii chisel [Elements of number theory]. Moscow: ONTI, 1936. Translated from the 6th Russian ed. by Helen Popova as An Introduction to the Theory of Numbers. London: Pergamon, 1955.

Metod trigonometricheskikh summ v teorii chisel. Moscow: Academy of Science of the USSR, 1947 (also Moscow: Nauka, 1971). Translated as The Method of Trigonometrical Sums in the Theory of Numbers. New York: Interscience, 1954 (also Mineola, NY: Dover, 2004).

Izbrannye trudy [Selected works]. Edited by Yu. V. Linnik. Moscow: Izd-vo Akademii nauk SSSR [Editorial house of the Academy of Sciences of the USSR], 1952.

Osobye varianty metoda trigonometricheskikh summ [Special variants of the method of trigonometric sums]. Moscow: Nauka, 1976.

Selected Works. Edited by L. D. Faddeev, R. V. Gamkrelidze, A.

A. Karacuba, et al. Berlin: Springer-Verlag, 1985. Pages 393–401 give a chronological list of Vinogradov’s works.

OTHER SOURCES

Karacuba, A. A. “Ivan Matveevich Vinogradov (k 90-letiyu so dnya rozhdeniya).” Uspekhi Matematicheskikh Nauk 36, no. 6 (1981): 3–15.

Mardzhanishvili, K. K. “Ivan Matveevič Vinogradov.” In Selected Works, edited by L. D. Faddeev, R. V. Gamkrelidze, A. A. Karacuba, et al. Berlin: Springer-Verlag, 1985.

S. Demidov