Ivan Matveevich Vinogradov ForMemRS(Russian: Ива́н Матве́евич Виногра́дов; 14 September 1891 – 20 March 1983) (not to be confused with Askold Ivanovich Vinogradov of the Bombieri-Vinogradov theorem) was a Soviet 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 he 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 Sergei Sobolev. In 1941 he was awarded the Stalin Prize.
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
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.