伊万·维诺格拉多夫Ivan Vinogradov

伊万·维诺格拉多夫的母亲是一名教师，他的父亲Matvei Avraam'evich 维诺格拉多夫是俄罗斯普斯科夫省大卢基区米洛柳布村的牧师。牧师之子通常接受特殊教会学校的教育，但由于维诺格拉多夫展现出科学才能，他的父母于1903年决定送他去学校接受科学而非古典教育。到1903年，维诺格拉多夫的父亲是大卢基圣裹尸布教堂的牧师，正是在那个城镇，维诺格拉多夫从1903年到1910年上学。

他在圣彼得堡大学学习，1910年进入那里的数学物理系。他在那里的两位老师马尔可夫和Ya V Uspenskii都对probability和数论感兴趣，维诺格拉多夫对数论的兴趣正是源于这一时期。他于1914年获得第一个学位毕业，由于他在quadratic residues和非剩余分布方面的杰出工作，他被鼓励在Uspenskii的指导下继续深造。

他于1915年完成硕士学位（见 [12] 和 [13]）：-

在他成功地为考试大纲非常广泛的硕士考试作准备的同时，维诺格拉多夫 正在研究数论中非常困难的问题……

他继续研究二次剩余，并于1915年经 弗拉基米尔·安德烈耶维奇·斯捷克洛夫 推荐获得奖学金。他推广了 格奥尔吉·沃罗诺伊 关于 约翰·彼得·古斯塔夫·勒热纳·狄利克雷 除数问题的结果，这使他能够对给定曲线 $y = f (x)$ 与 $x$ 轴之间的整点数目作出估计。

维诺格拉多夫在数学方法上非常专注，尽管首先面临第一次世界大战带来的困难，随后又面临俄国革命造成的动荡，他仍成功地推进了深入研究。尽管他尽了最大努力，当然仍有一些他无法克服的问题。主要涉及俄罗斯与西方之间缺乏交流，因此他不知道赫尔曼·外尔等人的结果，而这些结果与他的工作高度相关；同样，西方数学家也大多不知道维诺格拉多夫的结果。

1918年至1920年，他在彼尔姆国立大学任教。彼尔姆国立大学成立于1916年，曾一度称为莫洛托夫大学，现为高尔基国立大学。他的第一个职位是讲师（Docent），但一年后晋升为教授。1920年，他回到圣彼得堡担任两个职位，一个是理工学院教授，另一个是大学讲师。他在大学讲授数论课程，这门课程后来成为他关于该主题的著名教材Foundations of number theory的基础。1925年，他晋升为大学教授，成为概率论和数论部门的负责人。

大约从1930年起，他开始大量参与国家层面的数学管理工作，但他的研究工作却惊人地未受繁重工作量的影响。大约在这个时候，他为在列宁格勒的苏联科学院成立弗拉基米尔·安德烈耶维奇·斯捷克洛夫数学研究所做了所有的组织基础工作。1934年，他成为弗拉基米尔·安德烈耶维奇·斯捷克洛夫研究所的第一任所长，当科学院将研究所迁至莫斯科时，他也迁往莫斯科，并继续担任所长直至去世。作为这一时期他研究活动的一个指标，值得注意的是，他在1934年至1938年间每年发表约12篇论文。他在繁重的行政职责之上还能做到这一点，实在是非常了不起。

在担任弗拉基米尔·安德烈耶维奇·斯捷克洛夫研究所所长期间，维诺格拉多夫与尼古拉·卢津讨论了苏联数学研究所应重点研究的研究领域。这些领域是：分析与数学物理的基本问题；实变函数论的特殊领域；数论与伽罗瓦理论；概率论；理论力学；分析的应用方法。

现在我们稍微看一下维诺格拉多夫所做的主要数学贡献。三角和在数论中的重要性最早由赫尔曼·外尔在1916年表明。在1920年代，戈弗雷·哈罗德·哈代和利特尔伍德的工作发展了赫尔曼·外尔的方法，以攻击解析数论中的其他问题。然而，是维诺格拉多夫在1930年代的一系列论文中，将这一方法发挥到了极致。他的方法在1937年写成的Some theorems concerning the theory of 素数中达到了顶峰，该文为Goldbach conjecture提供了部分解决。在其中，维诺格拉多夫证明了每个充分大的奇数整数都可以表示为三个素数之和。在[8]中（或[9]，因为文章[8]和[9]是相同的），作者们写道：-

他引入并发展了两个基本方法，可以简要描述为“双线性形式技巧”和“均值定理”。它们使得在一系列问题上取得了进展。例如，在他可能是最著名的作品[关于素数理论的某些定理（1937）]中，他能够将双线性形式技巧与戈弗雷·哈罗德·哈代-利特尔伍德方法结合起来，从而将克里斯蒂安·哥德巴赫三元问题简化为检查有限个情况。

最近对维诺格拉多夫所研究类型问题的研究表明，他的方法仍然是获得进一步结果的最强大可用方法。

维诺格拉多夫做出了许多其他贡献，例如对幂剩余、非剩余、指数和原根的分布理论。他经常回到他第一篇研究论文的主题，即卡尔·弗里德里希·高斯发现的渐近公式中的误差项。书[2]包含维诺格拉多夫亲自选出的十六篇文章，他认为这些是最重要的。包括的论文有：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 爱德华·华林的theorem（1928）；以及Representation of an odd number as a sum of three primes（1937）。他的两部专著method of trigonometric sums in the theory of numbers和Special variants of the method of trigonometric sums也在书中。

他在苏联以外的影响很快就显现出来。即使在埃德蒙·朗道1927年出版的三卷本数论著作中，维诺格拉多夫的方法也受到了突出重视。然而，他很少出国旅行，尽管他确实在1958年作为苏联代表团团长访问了圣安德鲁斯，参加国际数学联盟（IMU）。随后他又前往爱丁堡参加国际大会。IMU前主席Chandrasekharan写道（见[8]或[9]）：-

维诺格拉多夫率领苏联代表团参加了在圣安德鲁斯举行的I.M.U.第三次大会，该大会恰好在1958年爱丁堡国际数学家大会之前举行。……维诺格拉多夫在1966年杜布纳举行的第五次大会上再次率领苏联代表团。

他和我过去常用英语交谈。我们曾长时间会面，有时讨论数学和数学家，有时谈论其他事情。我理解他的英语没有困难，他的回应表明他理解我说的话。我知道他同样能阅读和理解德语，尽管他从未特别想讲德语。

他确实欢迎来莫斯科拜访他的数学家。其中一位来访者⟦N1⟧写道：-

他是一位出色而细致的主人。……凡曾作为客人到他家做客的人，都不会忘记他慷慨的款待。

然而，24（另见25）的作者批评维诺格拉多夫，说他是一个典型的独来独往者，性格难以相处，只对他的数学研究和加强他的研究所感兴趣。

为庆祝他的80岁生日，在莫斯科举行了一次国际会议。维诺格拉多夫自费为与会者举办晚宴，并亲自填写邀请卡。会议论文集于1973年出版，维诺格拉多夫担任主编。

维诺格拉多夫因其数学成就获得了许多荣誉。他获得了苏联科学院所能授予的最高荣誉，即罗蒙诺索夫金质奖章。他还获得了许多其他苏联荣誉，例如：两次苏联英雄称号；五次列宁勋章；两次锤子与镰刀勋章；十月革命勋章；斯大林奖；以及列宁奖。他于1942年当选为伦敦皇家学会院士，并于1939年当选为伦敦数学会院士。

他一直身体健康，并为自己的体能感到自豪，直到90岁出头仍保持健康和活跃。

在他诞辰一百周年之际，9月14日组织了一次解析数论会议，随后是“维诺格拉多夫讲座”。文章[26]总结了10个一小时的“维诺格拉多夫讲座”，这些讲座致力于数论和代数几何中的相关问题。

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 the 3rd General Assembly of the I.M.U. at St Andrews, which was held just before the International Congress of Mathematicians at Edinburgh in 1958. ... Vinogradov headed the Soviet delegation again at the 5th General Assembly at Dubna in 1966.

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 80th 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.