图兰·帕尔Pál Turán

图兰·帕尔的父母是Aranha Beck和Béla 帕尔。帕尔（或匈牙利语中的帕尔 帕尔）是长子，有两个兄弟和一个姐妹。这个家庭是犹太人，因此不得不在极其艰难的时期中生存下来，遭受歧视，然后是猛烈的反犹主义。帕尔是布达佩斯中学的一名才华出众的学生，在这个阶段就展现出了他非凡的数学才能。

帕尔进入布达佩斯的Pázmány 罗莎·培特大学时已经显示出他的研究潜力。埃尔德什写道[7]：-

我们第一次见面是在1930年9月的布达佩斯大学，随即发现我们对数论有着共同的兴趣。

1933年，帕尔获得了文凭，这使他具备了教授数学和科学的资格，他继续攻读博士学位。他的第一篇论文发表于1933年，接下来的两篇论文均发表于1934年，都非常重要。一篇是A problem in the elementary theory of numbers，发表在American Mathematical Monthly上。它的重要性在于这是帕尔与埃尔德什的首次合作。第二篇是On a theorem of Hardy and Ramanujan，发表在Journal of the London Mathematical Society上。重要的不是帕尔在这里证明的结果，因为他证明的是一个自1917年起就已知的结果，即几乎所有整数$n$都有渐近log log $n$个素因子。重要的是证明方法，尽管它没有使用概率论的术语，实际上却成为了概率数论的基础之一。

他的博士学位由费耶尔指导，帕尔于1935年获得学位。他的学位论文On the number of prime divisors of integers用匈牙利语写成，已于1934年发表，其中包含了他对上述戈弗雷·哈罗德·哈代和拉马努金定理的新证明。即使在这个早期阶段，他已经建立了令人瞩目的国际声誉，到1935年底已有七篇论文付印，其中三篇发表在Journal of the London Mathematical Society上。人们可能会以为这位才华横溢的年轻数学家会轻易找到大学职位。然而，事实远非如此，因为对他犹太出身的严重歧视意味着他甚至无法获得学校教师的职位。为了在经济上支持自己，并给自己继续数学研究的机会，他不得不以私人数学家教为生。到1938年底，即他的第一篇论文发表五年后，他已在世界范围内国际重要期刊上发表了十六篇论文。他终于设法获得了学校教师的职位，1938年，他被任命为布达佩斯匈牙利拉比培训学校的数学助理教师。

1938年之所以重要，不仅因为帕尔至少有了工作，还因为这是他产生最富成果的数学想法的一年。埃尔德什在[7]中写道：-

帕尔最重要、最持久、最原创的成果大概要数他的幂和方法及其应用。1938年这个方法诞生时我就在场。帕尔提到了这些问题，并告诉我它们不仅本身有趣，而且其正面解决会有许多应用。它们的重要性首先在于它们引出了一类全新类型的有趣而深刻的问题；它们在数学的许多分支——微分方程、数值代数以及函数论的各个分支——中产生了完全出人意料的惊人结果。

事实上，帕尔是在研究ζ函数时发明了幂和方法，他首先用这个方法证明了关于zeta函数零点的结果。后来帕尔和S 斯塔尼斯瓦夫·克纳波夫斯基[3]：-

……研究了素数在模k的简化剩余类中的分布。……迄今为止，幂和方法被证明是研究这个问题的唯一方法。他们在这个领域的结果发表在近20篇论文中，被作者们称为比较数论。

如果说对帕尔来说，到1938年为止的岁月已经极其艰难，那么任何即将好转的迹象都是短暂的，因为很快情况就变得更糟了。帕尔在第一次世界大战期间长大，那是一个极为困苦的时期。战争结束后，1921年的《特里亚农条约》使匈牙利的领土缩减到此前的大约三分之一。随着事态向第二次世界大战发展，匈牙利的外交政策将德国和意大利视为能够帮助他们收复失地的盟友。在德国入侵波兰、第二次世界大战爆发后，匈牙利起初并未卷入，但仍深受纳粹政策的严重影响。1940年，帕尔被送往劳改营，整个战争期间他不断进出各种强制劳动营。这是一段可怕的经历，但正如我们下面将要指出的，也许最终正是由于这段经历，他的生命得以保全。Alpár写道[3]：——

但即使是[劳改营]也无法阻止他的数学活动。在任何情况下，他都利用最小的机会，在没有书籍和期刊、缺少同事陪伴的条件下继续他的研究，把想法和结果草草记在纸片上。他的若干新想法、问题以及如今著名的定理，都源自那个时期。正如G Alexits关于他所写的：“当法西斯暴行迫使他爬上电线杆拉电线时，他以思考数学问题来抵御这种恶意的压迫。有一次他告诉我：‘我最好的想法都是在拉电线时产生的，因为那时我可以独自一人，没有人注意到我在思考。’”

埃尔德什从1934年起开始与帕尔定期通信，最初能够在劳改营中与他取得一些联系。埃尔德什写信给在布达佩斯的父亲Lajos Erdős，后者再写信给帕尔，并抄录儿子信件中的相关部分。值得注意的是，其中一些信件甚至保存了下来，其英译文收录在[20]中，但1941年6月至1945年春之间没有任何通信记录。我们注意到，[20]的作者Vera T Sós是帕尔的妻子，他与她合写了许多论文。另一个值得注意的事实是，极值图论——帕尔所创立的一个领域——正是他在劳改营中拥有的“最好的想法”之一。

1941年，德国进攻俄罗斯，匈牙利支持他们。在俄国的抵抗远超预期之后，匈牙利动员了全部力量支持德国对俄罗斯的攻势。1943年1月，匈牙利军队在俄罗斯西部的沃罗涅日遭受了毁灭性的失败。1944年3月，匈牙利全面配合纳粹的目标，犹太人被迫佩戴黄星，被剥夺财产，并像其他纳粹占领区一样被强制迁入隔都。除了像帕尔那样在强迫劳动营中的犹太人外，其他人被送往德国集中营的毒气室。帕尔的两个兄弟和妹妹都在战争中丧生。据估计，匈牙利75万犹太人中约有55万在战争期间被杀害。帕尔于1944年从劳动营获释，并得以在布达佩斯的匈牙利拉比培训学校恢复教学。

第二次世界大战结束后，帕尔被任命为布达佩斯厄特沃什·罗兰大学的Privatdozent（该大学以前称为布达佩斯帕兹马尼培特大学）。匈牙利於1947年2月10日在巴黎签署了一项新的和平条约，恢复了特里亚农边界。然而在此之前，帕尔得以建立国际联系，使他在1947年访问丹麦六个月，随后在普林斯顿高等爱德华·斯图迪研究所六个月。回到匈牙利后，他于1948年当选为匈牙利科学院，并于同年获得匈牙利政府颁发的科苏特奖。1949年，他被任命为布达佩斯厄特沃什·罗兰大学的代数与数论讲席，直至去世。从1955年起，他担任匈牙利科学院数学研究所复变函数理论系主任。

埃尔德什在[7]中描述了他去世前不久的事件：——

……1976年7月，在巴黎奥赛举行的组合数学会议上，V T Sós（帕尔夫人）告诉了我那个可怕的消息（她已经知道六年了）：帕尔患了白血病。她告诉我应该尽快去看望他，并且和他交谈时要小心，因为他不知道自己病情的真实性质。我的第一反应是说也许应该告诉他……她说帕尔太热爱生活了，如果头上悬着死刑判决，就无法很好地生活和工作。……我现在相当确信她的决定是正确的，因为他显然从未试图查明自己病情的真实性质。事实上，在他去世前几天，[他的妻子]和他们的儿子George（也是一位数学家）试图说服他向Halász或Pintz口述书中的某些部分。他拒绝了，说“等我感觉好些、强壮些时我会写的”。不幸的是，他再也没有机会了。幸运的是，他的书由他的学生G Halász和J Pintz完成了……

这里提到的书是On a new method of analysis and its applications，于1984年出版。Bob Odoni写了一篇评论：-

1953年，作者出版了一本书《一种新的分析方法及其应用……》，系统阐述了他估计“幂和”的方法，他在1941年至1953年间将其发展成为一种多才多艺且强大的技术，在丢番图逼近、波恩哈德·黎曼 zeta函数的无零点区域和素数定理中的误差项以及经典分析其他部分的问题中有众多应用。关于后者，帕尔找到了诸如准解析类、Fabry间隙定理和缺项级数理论等主题的新方法。该书经过修订（改进了估计）出了第二版，但由于只有中文版，数学读者有限。1959年，帕尔开始准备该书的一个大幅扩充的新版本。由于新的改进和应用，不断需要重写，到1976年他去世时，这个项目仍未达到帕尔完全满意的程度。所评论的这本书代表了所有这些工作的顶峰……

Odoni在评论结尾对帕尔的数学致以这样的敬意：-

评论者认为，这本书为在经典分析广泛领域工作的数学家，特别是analytic number theorists，提供了巨大的帮助；帕尔的方法在当前研究中仍然具有重大意义，特别令人欣慰的是所有这些材料都收录在一卷之中。这本书是对帕尔在分析方面卓越成就的恰当致敬，手稿的编辑们为将其出版所做的努力值得高度赞扬。

我们在上面提到了帕尔的一些数学工作。然而，不可能公正地评价他所做的巨量工作，他发表了大约150篇论文。不过，我们提到他在统计group theory方面的工作，其中大部分是与埃尔德什合作进行的。当然，$S_{n}$在$n$个字母上的对称群的共轭类由$n$的分拆来刻画，因此与数论的联系是清楚的。帕尔和埃尔德什在这个主题上讨论的大多数问题涉及对称群$S_{n}$的随机元素的阶的分布。在他们考虑的一些问题中，所有排列被取为等概率的，另一些则是关于共轭类集合，全部等概率。帕尔和埃尔德什还证明了在阶为$n$的群中，至少有$n \log \log n$对$n^{2}$个元素对是可交换的。

一位在布达佩斯曾在帕尔手下工作的数学家这样描述他：-

在解析数论方面很出色，但不擅长管理一个系。

然而他为匈牙利科学院做了出色的工作，在众多委员会中任职。他还以多种方式为亚诺什·波尔约数学会服务，包括一度担任主席。帕尔的另一项重大贡献是编辑雷尼·奥尔弗雷德和费耶尔的论文集，这是Askey在文章[4]中提出的主要观点。Askey写道：-

我经常使用费耶尔的论文集。帕尔的编辑工作非常出色。他对许多论文作了评论，将它们置于背景之中，并说明费耶尔引入的思想后来发生了什么。

但这只是帕尔承担的编辑工作的一部分，他还担任Acta Arithmetica、Archiv für Mathematik、Analysis Mathematica、Compositio Mathematica、Journal of Number Theory以及基本上所有匈牙利数学期刊的编委。

除了我们上面提到的荣誉之外，帕尔还获得了许多荣誉。1952年，他第二次获得匈牙利政府颁发的科苏特奖。1975年，他还因创建科学学派而获得亚诺什·波尔约数学会颁发的Szele奖。他还当选为美国数学会、Austrian Mathematical Society和波兰数学会的成员。

1980年出版了一期献给帕尔的Acta Mathematica特刊。

Paul Turán's parents were Aranha Beck and Béla Turán. Paul Turán (or Turán Pál in Hungarian) was the eldest son having two brothers and one sister. The family were Jewish and so had to survive through exceedingly difficult times, suffering discrimination and then violent anti-Semitism. Paul was a brilliant pupil at secondary school in Budapest, showing at this stage his remarkable mathematical abilities.

Turán entered Pázmány Péter University of Budapest already showing his potential for research. Erdős writes [7]:-

We first met at the University of Budapest in September 1930 and immediately discovered our common interest in number theory.

In 1933 Turán was awarded his diploma which qualified him to teach mathematics and science, and he continued working for his doctorate. His first paper was published in 1933 and his next two papers, both published in 1934, were very significant. One was A problem in the elementary theory of numbers which appeared in the American Mathematical Monthly. It was significant in being Turán's first joint work with Erdős. The second was On a theorem of Hardy and Ramanujan which was published in the Journal of the London Mathematical Society. It was not the result which Turán proved here that was significant, for he proved a result which had been known since 1917, namely that almost all integers $n$ have asymptotically log log $n$ prime factors. Rather it was the method of proof which, although it does not use probabilistic terminology, in fact became one of the foundations of probabilistic number theory.

His Ph.D. was supervised by Fejér, and Turán was awarded the degree in 1935. His thesis On the number of prime divisors of integers, written in Hungarian, had been published in 1934 and contained his new proof of the theorem of Hardy and Ramanujan referred to above. Even at this early stage he had built up an impressive international reputation and had seven papers in print by the end of 1935, three of which had appeared in the Journal of the London Mathematical Society. One might have expected that this brilliant young mathematician would have easily found a university position. However, this was far from the case since the severe discrimination against him because of his Jewish origins meant that he could not even obtain a post as a school teacher. In order to support himself financially, and give himself the chance to continue his mathematical researches, he had to make a living as a private mathematics tutor. By the end of 1938, five years after his first paper appeared, he had sixteen papers in print in internationally important journals world-wide. At last he managed to get a position as a school teacher when, in 1938, he was appointed as an assistant teacher of mathematics at the Hungarian Rabbinical Training School in Budapest.

Not only was 1938 significant in that Turán now at least had employment, but it was also the year in which he had his most fruitful mathematical idea. Erdős writes in [7]:-

Probably the most important, most enduring and most original of Turán's results are in his power sum method and its applications. I was there when it originated in 1938. Turán mentioned these problems and told me that they were not only interesting in themselves but their positive solution would have many applications. Their importance first of all is that they lead to interesting deep problems of a completely new type; they have quite unexpectedly surprising consequences in many branches of mathematics - differential equations, numerical algebra, and various branches of function theory.

In fact Turán invented the power sum method while investigating the zeta function and he first used the method to prove results about the zeros of the zeta function. Later Turán and S Knapowski [3]:-

... investigated the distribution of primes in the reduced residue classes mod k. ... The power sum method proved to be the unique procedure for investigating this problem up to now. Their results in this field were published in nearly 20 papers and were called comparative number theory by the authors.

If times had been extremely hard for Turán up to 1938, then any appearance that they were about to get better was short lived for soon they became far worse. Turán had grown up during the years of World War I which had proved a time of great hardship. After the war ended, the Treaty of Trianon of 1921 saw Hungary's territory reduced to about one third of its previous size. As events moved towards World War II, Hungarian foreign policy looked towards Germany and Italy as allies who could help them to restore their lost territory. After the German invasion of Poland which began World War II, Hungary was not involved at first but was still greatly influenced by Nazi policies. In 1940 Turán was sent to a labour camp, and he was in and out of various forced labour camps throughout the war. This proved an horrific experience but, as we remark below, perhaps in the end his life was saved because of it. Alpár writes [3]:-

But not even [the labour camp] could stop his mathematical activity. In every situation, making use of the smallest opportunity, he carried on his research without books and journals, missing the company of colleagues, jotting down his ideas and results on scraps of paper. Several of his new ideas, problems and now famous theorems, originate from that period. As G Alexits has written about him: "When the fascist barbarism forced him to pull electric wires on poles he defended himself against the malevolent oppression be dealing with his mathematical ideas. Once he told me: "I got my best ideas while pulling wires, because then I could be alone and nobody noticed that I was thinking."

Erdős, who had begun corresponding regularly with Turán from 1934, initially was able to get some contact with him in the labour camps. Erdős wrote to his father, Lajos Erdős, in Budapest, who then wrote to Turán, copying out the relevant parts of his son's letters. Remarkably, even some of these letters have survived and they are reproduced in English translation in [20], but there is no record of any correspondence between June 1941 and Spring 1945. We note that Vera T Sós, the author of [20], was Turán's wife and he wrote a number of joint papers with her. Another remarkable fact is that extremal graph theory, an area which Turán founded, was one of the "best ideas" that he had while in the labour camps.

In 1941 Germany attacked Russia and Hungary supported them. After the Russian resistance was far greater than expected, Hungary mobilised all its forces to support the German offensive on Russia. The Hungarian forces suffered a crushing defeat at Voronezh in western Russia in January 1943. In March 1944 Hungary fully cooperated with Nazi aims and Jews were forced to wear a yellow star, robbed of their property, and forced into ghettos as in other Nazi-occupied areas. Except for the Jews in the forced-labour camps, like Turán, others were sent to the gas chambers of German concentration camps. Turán's two brothers and his sister all died during the war. It is estimated that 550,000 of Hungary's 750,000 Jews were killed during the war. Turán was liberated from the labour camp in 1944 and was able to resume teaching at the Hungarian Rabbinical Training School in Budapest.

After World War II ended Turán was appointed as a Privatdozent at the Eötvös Loránd University of Budapest (it had formerly been called the Pázmány Péter University of Budapest). Hungary signed a new peace treaty in Paris on 10 February 1947, which restored the Trianon frontiers. Before this, however, Turán was able to make international contacts which let him visit Denmark for six months, then the Institute for Advanced Study at Princeton for six months, in 1947. On his return to Hungary he was elected to the Hungarian Academy of Sciences in 1948, and received the Kossuth Prize from the Hungarian government in the same year. In 1949 he was appointed to the Chair of Algebra and Number Theory at Eötvös Loránd University of Budapest, a position he held until his death. From 1955 he was Head of the Complex Function Theory Department in the Mathematical Institute of the Hungarian Academy of Sciences.

Erdős in [7] describes events just before his death:-

... in July 1976, at the meeting on combinatorics at Orsay in Paris, V T Sós (Mrs Turán) gave me the terrible news (which she had known for six years) that Paul had leukaemia. She told me that I should visit him as soon as possible and that I should be careful in talking to him because he did not know the true nature of his illness. My first reaction was to say that perhaps he should have been told ... She said that Paul loved life too much and with a death sentence hanging over him would not be able to live and work very well. ... I am now fairly sure that her decision was right, since he clearly never tried to find out the true nature of his illness. in fact a few days before his death [his wife] and their son George (also a mathematician) tried to persuade him to dictate some parts of his book to Halász or Pintz. he refused saying "I will write it when I feel better and stronger". Unfortunately he never had the chance. Fortunately his book was finished by his students G Halász and J Pintz ...

The book mentioned here is On a new method of analysis and its applications which was published in 1984. Bob Odoni wrote a review:-

In 1953 the author published a book, A new method of analysis and its applications ... giving a systematic account of his methods for estimating "power sums", which he had developed (1941-53) into a versatile and powerful technique with numerous applications to Diophantine approximations, zero-free regions for the Riemann zeta function and the error term in the prime number theorem, and to problems in other parts of classical analysis. As regards the latter, Turán found new approaches to such topics as quasi-analytic classes, Fabry's gap theorem and the theory of lacunary series, amongst others. The book was revised (with improved estimates) in a second edition, but this had a limited mathematical audience since it was only available in Chinese. In 1959 Turán embarked on the preparation of a new, greatly expanded version of the book. Constant rewriting became necessary in the light of the new improvements and applications, and, at the time of his death in 1976, the project had still not been completed to Turán's total satisfaction. The book under review represents the culmination of all this work ...

Odoni ends his review with this tribute to Turán's mathematics:-

In the opinion of the reviewer this book renders a great service to mathematicians working in a wide area of classical analysis, particularly analytic number theorists; Turán's methods are still of great relevance in current research, and it is particularly gratifying to have all this material within the confines of a single volume. The book is a fitting tribute to Turán's remarkable achievements in analysis, and the editors of the manuscript deserve high praise for their efforts in bringing it to publication.

We have mentioned some of Turán's mathematics above. However, it is impossible to do justice to the huge amount of work which he did, publishing around 150 papers. We mention, however, his work on statistical group theory, much of which was undertaken jointly with Erdős. Of course conjugacy classes of the symmetric group $S_{n}$ on $n$ letters are characterized by partitions of $n$, so the connection with number theory is clear. Most questions discussed by Turán and Erdős on this topic concern the distribution of the order of random elements of the symmetric group $S_{n}$ . In some of the problems they considered, all permutations are taken to be equally probable, some others are about the set of conjugacy classes, all equally probable. Turán and Erdős also proved that in a group of order $n$, at least $n \log \log n$ of the $n^{2}$ pairs of elements commute.

A mathematician who served under Turán in Budapest described him as:-

... outstanding in analytic number theory but not a good manager of a department.

However he did outstanding work for both the Hungarian Academy of Sciences, serving on numerous committees. He also served the János Bolyai Mathematical Society in many ways including a time as president. Another major contribution made by Turán was his editing of the papers of Rényi and Fejér which is the main point made by Askey in the article [4]. Askey writes:-

I have used the Fejér papers often. Turán's editing was remarkable. He commented on many of the papers, setting them in context and telling what happened to the ideas Fejér introduced.

But this is only a part of the editorial work Turán undertook, being on the editorial boards of Acta Arithmetica, Archiv für Mathematik, Analysis Mathematica, Compositio Mathematica, Journal of Number Theory, and essentially all Hungarian mathematical journals.

Turán received many honours in addition to the honours which we mentioned above. He received the Kossuth Prize from the Hungarian government for a second time in 1952. He also received the Szele Prize from the János Bolyai Mathematical Society in 1975 for creating scientific schools. He was also elected a member of the American Mathematical Society, the Austrian Mathematical Society, and the Polish Mathematical Society.

A special issue of Acta Mathematica devoted to Paul Turán was published in 1980.