Ivan Matveyevich Vinogradov

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Ivan Matveyevich Vinogradov
Ivan Matveyevich Vinogradov
Ivan Matveyevich Vinogradov
Born September 14, 1891
Died March 20, 1983
Nationality Russian
Fields mathematics
Known for analytic number theory

Ivan Matveevich Vinogradov (Иван Матвеевич Виноградов: September 14, 1891–March 20, 1983) was a Russian mathematician, who was one of the creators of modern analytic number theory, and also a dominant figure in mathematics in the USSR. He was born in the Velikiye Luki district, Pskov Oblast. He graduated from the University of St. Petersburg, where in 1920 became a Professor. From 1934 he was a Director of the Steklov Institute of Mathematics, a position he held for the rest of his life, except for the five-year period (1941–1946) when the institute was directed by Academician Sobolev.

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[edit] Mathematical contributions

In analytic number theory, Vinogradov's method refers to his main problem-solving technique, applied to central questions involving the estimation of exponential sums. In its most basic form, it is used to estimate sums over prime numbers, or Weyl sums. It is a reduction from a complicated sum to a number of smaller sums which are then simplified. The canonical form for prime number sums is

S=\sum_{p\le P}\exp(2\pi i f(p)).

With the help of this method, Vinogradov tackled questions such as the ternary Goldbach problem in 1937 (using Vinogradov's theorem), and the zero-free region for the Riemann zeta function. His own use of it was inimitable; in terms of later techniques, it is recognised as a prototype of the large sieve method in its application of bilinear forms, and also as an exploitation of combinatorial structure. In some cases his results resisted improvement for decades.

He also used this technique on the Dirichlet divisor problem, allowing him to estimate the number of integer points under an arbitrary curve. This was an improvement on the work of Georgy Voronoy.

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[edit] Political and institutional aspects

Vinogradov was a Communist Party official, not unusual for a highly placed administrator. Some have concluded from his prominence that he must have known of the repressive trends of the Soviet system, as they had an impact on the mathematical community of the USSR, continuing into the Brezhnev era.

It is a matter of dispute if Vinogradov himself was subject to particular pressure from the KGB and other elements of the Soviet system to implement such policies. Several prominent mathematicians have accused him of complicity, basing their judgment on their perception of his personal character and behavior and those of his associates.

According to Sergei Petrovich Novikov (S.P. Novikov. Rokhlin. Rokhlin. Retrieved on 2004-08-19.), Vinogradov began pursuing antisemitic moves in his career starting in 1950s, although having never been an antisemitist before, "while it had not been profitable", i.e., until antisemitism became a part of the Stalin terror after World War II. According to the same source, Vinogradov obstructed Jewish and dissident Soviet scientists by requesting them secretly investigated by the KGB, trying to prevent their career promotion, directly as well as by persuading other scientists, and preventing their voyages abroad (often by secretly reporting them as "untrustworthy" to the party functioneers and KGB officials).

[edit] Bibliography

  • Selected Works, Berlin ; New York : Springer-Verlag, 1985, ISBN 0-387-12788-7
  • Vinogradov, I.M. Elements of Number Theory. Mineola, NY: Dover Publications, 2003, ISBN 0-486-49530-2
  • Vinogradov, I.M. Method of Trigonometrical Sums in the Theory of Numbers. Mineola, NY: Dover Publications, 2004, ISBN 0-486-43878-3
  • Vinogradov I.M. (Ed.) Matematicheskaya entsiklopediya. Moscow: Sov. Entsiklopediya 1977. Now translated as the Encyclopaedia of Mathematics.

[edit] External links