# Ivan Matveevich Vinogradov

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## Vinogradov, Ivan Matveevich

Born Sept. 2 (14), 1891, in the village of Miroliub, now in Velikie Luki Raion, Pskov Oblast. Soviet mathematician. Academician of the Academy of Sciences of the USSR (1929). Hero of Socialist Labor (1945).

Vinogradov was born into the family of a village priest. Upon graduating from the University of St. Petersburg (1914) he was kept on at the university to prepare for the rank of professor. Between 1918 and 1920 he was a docent and a professor at the University of Perm’ and between 1920 and 1934 a professor at the Leningrad Polytechnic Institute and Leningrad University. In 1932 he became director of the V. A. Steklov Institute of Mathematics of the Academy of Sciences of the USSR. His works are devoted to the analytical theory of numbers. Vinogradov has solved problems considered inaccessible to the mathematics of the early 20th century. He originated the method of trigonometric sums, one of the most powerful and general methods of the analytical theory of numbers. Almost all problems of the analytical theory of numbers are rather simply formulated in the language of finite sums of terms of the form cos *F(X _{1},....,x_{n}*) + i sin

*F*(X

_{1},....., x

_{n}), where

*F*is a real integral function and i = √-1. Thus, the main task in many problems becomes a matter of studying such sums and, in particular, of obtaining as exact an estimate for the module of such sums as possible. By taking advantage of the profound arithmetic properties of the sums studied, Vinogradov obtained unusually powerful estimates of the module of a broad class of such sums. This method enabled Vinogradov to obtain the best possible basic results for a whole series of problems of the theory of numbers in such classic problems as Waring’s problem, the Hilbert-Kamke problem, and Weyl’s problem of the evaluation of sums (1934-35). Another result of the method (1935-37) was the solution of a number of additive problems with simple numbers and, in particular, the solution of Goldbach’s problem. Vinogradov’s methods have become standard and are being developed and used by many scientists in various branches of mathematics. Vinogradov is an honorary member of the London Royal Society (1942) and many other scientific institutions and societies of the world. He has been awarded the M. V. Lomonosov Gold Medal (1971), the State Prize of the USSR (1941), four Orders of Lenin, and various medals.

### WORKS

*Izbr. trudy*. Moscow, 1952.

*Osnovy teorii chisel*, 7th ed. Moscow, 1965.

*Metod trigonometricheskikh summ v teorii chisel*. Moscow, 1971.

### REFERENCE

Linnik, Iu. V., and A. G. Postnikov. “I. M. Vinogradov (k 70-letiiu so dnia rozhdeniia).”*Uspekhi matematicheskikh nauk*, 1962, vol. 17, issue 2 (104), pp. 201-214. (Contains a listing of Vinogradov’s works.)