**Ivan Matveevich Vinogradov**'s mother was a teacher and his father, Matvei Avraam'evich Vinogradov, was priest in Milolyub, a village in the Velikie Luki district of the Pskov province of Russia. The usual education for the son of a priest was in a special ecclesiastic school, but because Ivan showed scientific skills his parents decided in 1903 to send him to school were he would receive a scientific rather than classical education. By 1903 Ivan's father was a priest at the Church of the Holy Shroud in Velikie Luki, and it was in that town that Ivan attended school from 1903 to 1910.

He studied at the university in St Petersburg, entering the Faculty of Mathematics and Physics there in 1910. Two of his teachers there, A A Markov and Ya V Uspenskii, both had interests in probability and number theory and Vinogradov's interest in number theory stems from this period. He graduated with his first degree in 1914 and, because of his outstanding work on the distribution of quadratic residues and non-residues, he was encouraged to continue his studies, supervised by Uspenskii.

His master's degree was completed in 1915 (see [12] and [13]):-

While he was successfully preparing for the Master's examination with its very broad syllabus, Vinogradov was working on very difficult problems in the theory of numbers ...

He continued to work on quadratic residues, having been awarded a scholarship in 1915 on the recommendation of Steklov. He generalised results of Voronoy on the Dirichlet divisor problem which allowed him to obtain estimates for the number of integral points between a given curve *y* = *f* (*x*) and the *x*-axis.

Vinogradov was very single minded in his approach to mathematics and succeeded to press ahead with deep research despite the difficulties arising first from World War I, and then from the upheaval caused by the Russian revolution. Despite his best efforts, there were, of course, problems which he could not overcome. Mainly these concerned the lack of communication between Russia and the West so that he was unaware of results of Weyl and others which was highly relevant to his work and, similarly, mathematicians in the West were largely unaware of Vinogradov's results.

He taught at the State University of Perm from 1918 to 1920. The State University of Perm, founded in 1916, was called Molotov University for a time, and is now the Gorky State University. His first appointment was as a docent, but after a year he was promoted to professor. In 1920 he returned to St Petersburg to two posts, one as professor at the Polytechnic Institute, and the other as docent 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 *Foundations of number theory.* 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. Around this time he did all the organizational groundwork for the foundation of the Steklov Mathematical Institute at the USSR Academy of Sciences in Leningrad. He become the first director of the Steklov Institute in 1934, moving to Moscow when the Academy moved the Institute there, and continuing to hold the directorship 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. How he achieved this on top of his heavy administrative duties is quite remarkable.

During his time as head of the Steklov Institute, Vinogradov discussed with Luzin the research areas which should be emphasised in the Soviet mathematical Institutes. These were: fundamental questions of analysis and mathematical physics; special areas of function theory of real variables; number theory and Galois theory; probability theory; theoretical mechanics; applied methods of analysis.

Let us now look a little at the main mathematical contributions made by Vinogradov. The importance of trigonometric sums in the theory of numbers was first shown by Weyl in 1916. In the 1920s 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 1930s, 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 primes. In [8] (or [9] for the articles [8] and [9] are identical) 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.

Recent research on the type of problems studied by Vinogradov shows that his methods are still the most powerful available to obtain yet further results.

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. The book [2] contains sixteen articles by Vinogradov which he selected himself as those he felt were most significant. The papers included: *On the distribution of power residues and nonresidues* (1918); *On the distribution of fractional parts of values of a function of one variable* (1926); *On* Waring's *theorem* (1928); and *Representation of an odd number as a sum of three primes* (1937). Two of his monographsThe *method of trigonometric sums in the theory of numbers,* and *Special variants of the method of trigonometric sums* are also in the book.

His influence outside the Soviet Union was soon apparent. Even in Edmund 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 (IMU). He then went on to the International Congress at Edinburgh. Chandrasekharan, a former president of the IMU wrote (see [8] or [9]):-

Vinogradov headed the U.S.S.R. delegation to the3rd General Assembly of the I.M.U. at St Andrews, which was held just before the International Congress of Mathematicians at Edinburgh in1958. ... Vinogradov headed the Soviet delegation again at the5th General Assembly at Dubna in1966.

He and I used to converse in English. We have met for long hours, sometimes discussing mathematics and mathematicians, at other times about other things. I had no difficulty in understanding his English, and his responses showed that he understood what I said. I know that he could read and understand German just as well, though he never particularly wanted to speak German.

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

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

The authors of [24] (see also [25]), however, criticise Vinogradov saying that he was a typical loner with a difficult character, interested only in his mathematical research and in strengthening his Institute.

An international conference was held in Moscow to mark his 80^{th} birthday. Vinogradov gave a dinner for the participants at his own expense and personally addressed the invitation cards. The proceeding of the conference were published in 1973 with Vinogradov as editor-in-chief.

Vinogradov received many honours for his mathematical achievements. He received the highest honour the USSR Academy of Sciences could give, namely the Lomonosov Gold Medal. He also received many other Soviet honours such as: Hero of the Soviet Union, on two occasions; Order of Lenin, on five occasions; Order of the Hammer and Sickle, on two occasions; the Order of the October Revolution; The Stalin Prize; and the Lenin Prize. He was elected to the Royal Society of London in 1942 and to the London Mathematical Society in 1939.

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

One hundred years after his birth, on 14 September, a conference on analytic number theory was organised, followed by 'Vinogradov lectures'. The article [26] gives summaries of the 10 one hour 'Vinogradov lectures' devoted to number theory and related problems in algebraic geometry.

**Article by:** *J J O'Connor* and *E F Robertson*

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