Иван Матвеевич Виноградов

Ivan Vinogradov used trigonometric series to attack deep problems in analytic number theory.

His master's degree was completed in 1915 and he worked on quadratic residues. He generalised results of Voronoy on the Dirichlet divisor problem.

Vinogradov taught at the State University of Perm' from 1918 to 1920. The State University of Perm' had been founded in 1916, was called Molotov University for a time. He returned to St Petersburg to two posts, one at the polytechnic and the other at the university. He gave a course on number theory at the university which was to be the basis for his famous text on the subject. He was promoted to professor at the university in 1925, becoming head of the probability and number theory section.

From around 1930 he became heavily involved with mathematics administration on a national level but his research work was amazingly unaffected by the heavy workload. He moved to Moscow to become the first director of the Steklov Institute in 1934, a post he held until his death. As an indication of his research activity during this period it is worth noting that he published around 12 papers in each of the years 1934 to 1938.

The importance of trigonometric sums in the theory of numbers was first shown by Weyl in 1916. In the 1920's the work of Hardy and Littlewood developed Weyl's methods to attack other problems in analytic number theory. However it was Vinogradov who, in a series of papers in the 1930's, brought the method to its full potential.

His methods reached their height in Some theorems concerning the theory of prime numbers written in 1937 which provides a partial solution to the Goldbach conjecture. In it Vinogradov proved that every sufficiently large odd integer can be expressed as the sum of three odd primes. In [3] the authors write:-

He introduced and developed two fundamental methods, which could be briefly described as 'the bilinear form technique' and 'the mean value theorem'. They have enabled progress to be made on a whole range of problems. For example, in what is probably his most celebrated piece of work [Some theorems concerning the theory of prime numbers (1937)], he was able to combine the bilinear form technique with the Hardy-Littlewood method so as to reduce the Goldbach ternary problem to that of checking a finite number of cases.

Vinogradov made many other contributions, for example to the theory of distribution of power residues, non-residues, indices and primitive roots. He often returned to the topic of his first research paper on the error term in an asymptotic formula discovered by Gauss.

Vinogradov's influence outside the Soviet Union was soon apparent. Even in Landau's three volume work on number theory, published in 1927, prominence is given to Vinogradov's methods. However he seldom travelled outside the Soviet Union although he did visit St Andrews in 1958 as the leader of the Soviet delegation to the International Mathematical Union. He then went on to the International Congress at Edinburgh.

He did welcome mathematicians who visited him in Moscow. One such visitor wrote:-

He was a marvellous and meticulous host. ... No one who has been at his home as a guest can forget his bountiful hospitality.

Always a fit man, and proud of his physical fitness, he remained healthy and active into his early 90s.

Fellow of the Royal Society (since 1942).

List of References

Article by: J.J. O'Connor and E.F. Robertson
Source:       MacTutor History of Mathematics archive