雷尼·奥尔弗雷德Alfréd Rényi

雷尼·奥尔弗雷德的父母是Artur Rényi和Barbara Alexander。Artur Rényi是一位学识渊博的机械工程师和语言学家，Barbara Alexander是布达佩斯大学哲学与美学教授Bernát Alexander的女儿。奥尔弗雷德的父母都是犹太人，这一事实对于生活在这一反犹狂热时期的匈牙利人来说，不幸地具有极其重要的意义。他的祖父，即Artur的父亲，出生于德国，家族姓氏为Rosenthal。然而他离开了德国，在澳大利亚当过一段时间的牧羊人后，定居匈牙利。他将姓氏从Rosenthal改为奥尔弗雷德，并通过他创办的手杖工厂发家致富。奥尔弗雷德的一位叔父是著名精神分析学家Franz Gabriel Alexander。

奥尔弗雷德是他父母的独子，在布达佩斯的一所文理中学接受的是文学而非科学的教育。他一生热爱文学，并对古希腊哲学家的著作着迷。这或许部分继承自他的母亲，部分源于他的学校教育。他还开始对天文学产生兴趣，这种兴趣又引导他对物理学产生兴趣。为了理解物理学，他必须掌握一些困难的数学，很快他就清楚地认识到，这才是他应该在大学学习的正确科目。促使他转向数学的另一个因素，是他在中学里杰出的数学老师罗莎·培特。1939年，他以年级最优异的成绩从中学毕业，但由于他的犹太父母，匈牙利国家实施的种族法律使他无法进入布达佩斯大学学习。然而，他仍然能够展示自己杰出的能力，在1939年秋季的一次数学竞赛和一次希腊语竞赛中都获得了荣誉奖。在这个阶段，奥尔弗雷德开始在甘茨造船和起重机厂工作。他在那里做了大约六个月的劳工，直到1940年10月获准进入布达佩斯大学学习数学和物理学。他的本科授课教师包括费耶尔和图兰·帕尔，后者写道[19]：-

我第一次见到他是在1941年10月或11月，当时他……开始定期参加我举办的一系列讲座，一直持续到1942年6月。讨论的主要主题是数论……

1944年5月，奥尔弗雷德毕业于布达佩斯大学，但在接下来的一个月里，他被迫进入法西斯劳役营，但不知何故设法逃脱了[19]：-

幸运的是，他的连队没有立即被带离匈牙利。当向西撤离的命令真的到来时，奥尔弗雷德逃脱了，并使用假证件住在布达佩斯。

他获得了假证件并躲藏了六个月，设法避免被捕。在此期间，他的父母被关押在布达佩斯犹太人隔离区。奥尔弗雷德以极其勇敢的行为救出了他们[20]：-

奥尔弗雷德弄到一套士兵制服，走进隔离区，把他的父母带了出来。……需要熟悉当时的情况才能体会到完成这些壮举所需的技巧和勇气。

图兰·帕尔写道[19]：-

那些日子里每当我遇到他，我都对他的冷静和勇气感到惊讶。

1945年3月，第二次世界大战接近尾声时，奥尔弗雷德在塞格德大学获得博士学位，学位论文导师是里斯，论文主题是奥古斯丁·路易·柯西-约瑟夫·傅里叶级数。他博士论文的成果发表在论文On the summability of Cauchy-Fourier series（1950）中。获得博士学位后，他离开塞格德，回到布达佩斯，在那里从事各种统计工作。在布达佩斯期间，他于1946年与Katalin Schulhof（1924-1969）结婚。他们有一个孩子，女儿Zsuzsa生于1948年。Katalin（也被称为Catherine或Kató）也是一位数学家，写了21篇论文，其中一些与丈夫合作。除了从事分析研究外，她还非常成功地鼓励年轻大学生在数学各个分支的研究。János Bolyai Mathematical Society以她的名义颁发Kató 奥尔弗雷德纪念奖。

奥尔弗雷德 作为博士后前往俄罗斯，在1946年10月至1947年6月期间，与 尤里·弗拉基米罗维奇·林尼克 一起研究数论，特别是研究 Goldbach conjecture [19]：-

在那几个月里，他的发展简直是惊人的。凭借意志的努力，他抹去了对战争年代和强迫劳动营的记忆，现在将青春的全部炽热能量以及他非凡的理解力和专注力投入到工作中。

他发现的方法被 图兰·帕尔 描述为：-

……目前是 analytical number theory 最强的方法之一。

奥尔弗雷德 在俄罗斯获得的结果发表在论文 On the representation of an even number as the sum of a single prime and a single almost-prime number（俄文）（1947）中。L Schoenfeld 将这些结果总结如下：-

一组整数 $S$，具有这样的性质：存在一个绝对常数 $K$，使得 $S$ 中的每个 $x$ 至多有 $K$ 个不同的素因子，称为殆素数集。$S$ 中的每个 $x$ 称为殆素数。作者指出了证明，将另处详细给出，即每个偶数都是一个殆素数（取自一个固定集合 $S$）和一个素数之和。他还声明他能证明存在无穷多个素数 $p$，使得 $p + 2$ 是殆素数（属于一个固定集合 $S^{*}$）。第一个结果，关于偶数的表示，是对未证明的 Goldbach conjecture 的近似，并取代了 特奥多尔·埃斯特曼（1932）对同一命题的早期证明，该证明利用了对所有 约翰·彼得·古斯塔夫·勒热纳·狄利克雷 $L$-级数的未证明的广义 波恩哈德·黎曼 假设。第二个结果是对存在无穷多个 twin primes 的猜想的近似，显然是一个新结果。

第二年，奥尔弗雷德发表了这些结果的完整证明，并且指出，他的方法还表明每个奇数都是一个素数加上两倍一个殆素数的和，而且对于每个固定的整数m（正或负），都存在无穷多个素数$p$使得$p + m$是殆素数。

他早期职业生涯中发表的其他论文包括：On a Tauberian theorem of O Szász（1946）；Integral formulae in the theory of convex curves（1947）；On the minimal number of terms of the square of a polynomial（1947）；On some new applications of the method of Academician I M Vinogradov（1947）；（与尤里·弗拉基米罗维奇·林尼克）On certain hypotheses in the theory of Dirichlet characters（俄语）（1947）。回到匈牙利后，他于1947年10月被任命为布达佩斯大学的Privatdocent和助理教授。1949年至1950年10月期间，他是德布勒森大学的非常任教授。他于1949年当选为匈牙利科学院的通讯成员。1950年，他被任命为匈牙利科学院新应用数学研究所的所长。1952年，除了其他职务外，他还被任命为布达佩斯厄特沃什·罗兰大学概率与统计系的教授。

奥尔弗雷德研究概率论，这成为他毕生的主要研究课题，但他的兴趣广泛，还涵盖统计学、信息论、组合学、图论、数论和分析。他的出版物清单惊人地包含355项。大卫·乔治·肯德尔写道[1]：-

在像尤里·弗拉基米罗维奇·林尼克、埃尔德什和奥尔弗雷德这样的作者手中，数论与概率论并没有明确区分。两者相互借用技巧，重要问题位于它们的共同前沿。因此，当奥尔弗雷德被称为伟大的应用概率学家时，部分原因是他对概率应用于数学其他部分的兴趣。

他与埃尔德什合作发表了关于随机图的论文，最重要的是On the evolution of random graphs（1960），并且解决了关于随机空间填充曲线的一个突出猜想，见On a one-dimensional problem concerning random space filling（1958）。他还撰写了许多杰出的著作，包括The calculus of probabilities（匈牙利文）（1954）。Eugene Lukacs写道：-

概率论及其应用在匈牙利大学的课程中一直被忽视，直到最近作者开始定期就这些主题授课。因此，他不得不为学生准备油印笔记，经过反复修订，最终促成了本书的出版。因此，这是第一本现代匈牙利语概率论教科书，并提供了对该领域的出色介绍。……遗憾的是，这本书只对匈牙利学生有用；它值得被添加到匈牙利科学院的外文出版物中。

事实上，1962年出版了一本经过彻底修订和完全重组的德文版，书名为Wahrscheinlichkeitsrechnung. Mit einem Anhang über Informationstheorie。1966年出现了法文版，1970年出版了英文版，包含三个新章节，书名为Probability theory。另一本书也于1970年出版，是Foundations of probability。这本书基于条件概率空间的概念，发展了一种与之前出版的所有关于该主题的书完全不同的概率方法。这种方法基于奥尔弗雷德发明的一套新公理系统，并在1954年9月2日至9月9日于阿姆斯特丹举行的国际数学家大会上的一次演讲中提出。

他的许多朋友和同事都昵称他为“布巴”，他常被记得是下面这则轶事的作者：-

……数学家是把咖啡转化为定理的机器

图兰·帕尔将这则轶事进一步发挥，把淡咖啡描述为只配用来证明引理。奥尔弗雷德是一位著名的健谈者，人们记得他多次表演他那种向女儿讲述数学本质的对话。他以这种风格出版了Dialoge über Mathematik Ⓣ（《关于数学的对话》）（1967年）和Letters on probability（匈牙利文版1969年，英文译本1972年）。这部著作是对概率论某些早期思想的一次引人入胜的半通俗、半历史的叙述。奥尔弗雷德采用的表述风格是收集布莱兹·帕斯卡写给皮埃尔·德·费马的（虚构）信件。他假定回信已经遗失。为了让人领略Dialoge über MathematikⓉ（《关于数学的对话》）的风味，我们引用奥尔弗雷德让阿基米德对希耶罗国王[2]所说的话：-

数学就像你的女儿海伦娜，每当有求婚者出现，她就怀疑他并非真的爱她，而只是对她感兴趣，因为他想当国王的女婿。她想要一个因她自身的美貌、机智和魅力而爱她的丈夫，而不是为了娶她所能得到的财富和权力。同样，数学只向那些怀着纯粹的爱、为数学自身之美而接近它的人揭示自己的秘密。这样做的人当然也会得到具有实际重要性的成果作为回报。但如果有人在每一步都问“我能从中得到什么好处？”，他就走不远。

奥尔弗雷德因其成就获得过许多荣誉，倘若他不是在48岁的悲剧性年纪去世，无疑还会获得更多荣誉。他曾两次被匈牙利政府授予科苏特奖，当选为国际统计学会副主席，担任过亚诺什·波尔约数学会的秘书（1949—55年），并受邀担任八种期刊的编委。在他突然去世后，人们发现了他正在撰写的一本书Diary on information theory的材料。Gyula Katona利用奥尔弗雷德为该书其余部分所作的笔记完成了它，该书先以匈牙利文出版，然后于1982年以德文出版，1984年以英文出版，此时距奥尔弗雷德去世已有十四年。

大卫·乔治·肯德尔在[10]中让我们对奥尔弗雷德的工作方式有了多一点了解。他是：-

……一位在数论与分析这些经典领域中成就巨大、地位崇高的纯粹数学家。奥尔弗雷德对周围世界的一切现象，以及同事们的一切学术活动——无论是科学的还是人文的——都怀有好奇而执着的兴趣，这种能力与兴趣的独特结合，使他得以建立起一个研究所，在那里，一个课题是否可被接受加以研究的标准是“是否至少有一位数学家对该课题怀有真正的兴趣”？然而，一旦被认定为合适，该课题就会以彻底专业的方式加以推进；论证会循其逻辑所至而展开，并以任何看来合适的数学手段加以支撑，无论这些手段多么奇特或高深。

最后，让我们引用图兰·帕尔关于奥尔弗雷德闲暇兴趣的记述[19]：-

奥尔弗雷德对生活的热情，并未在他多方面的智力活动和公共事务参与中耗尽。他喜欢夏天在多瑙河上划船和游泳，冬天则喜欢滑雪。他与妻子和女儿Zsuzsa常去听音乐会、看戏；在他们家中举办的聚会上，奥尔弗雷德以风趣的轶事和弹钢琴来款待朋友。进入他书房的人会看到书架一直塞到天花板。书桌上散乱堆放的书、手稿和笔记，让来访者感到自己进入了创造性、生产性活动的现场，而奥尔弗雷德直到生命的最后一天都未曾减弱地从事着这种活动。

Alfréd Rényi's parents were Artur Rényi, a mechanical engineer and linguist of wide learning, and Barbara Alexander, the daughter of Bernát Alexander, the professor of philosophy and aesthetics at Budapest University. Both of Alfréd's parents were Jewish, a fact which, sadly, was highly significant for those living in Hungary through this period of anti-Semitic fervour. His paternal grandfather, Artur's father, was born in Germany with the family name of Rosenthal. However he left Germany and, after spending a while as a sheep farmer in Australia, settled in Hungary. He changed his name from Rosenthal to Rényi and made his fortune from a walking stick factory which he founded. One of Alfréd's uncles was Franz Gabriel Alexander, the famous psychoanalyst.

Alfréd, his parents' only child, received a literary, rather than scientific, schooling at a Gymnasium in Budapest. Throughout his life he had a love of literature and was fascinated by the works of the philosophers of ancient Greece. This, perhaps, was partly inherited from his mother and partly a result of his school education. He also began to take an interest in astronomy and this interest led him to take an interest in physics. To understand physics he had to master some difficult mathematics and it soon became clear to him that this was the right subject for him to study at university. Another factor in his move towards mathematics was his outstanding mathematics teacher at the Gymnasium, Rózsa Péter. He graduated from the Gymnasium in 1939 as the best student in his year but, because of his Jewish parents, he was unable to study at Budapest University due to the racial laws imposed by the Hungarian state. However, he was still able to show his outstanding abilities by receiving an honorable mention in both a mathematics competition and in a Greek competition in the autumn of 1939. At this stage Rényi began work at the Ganz Shipyard and Crane Factory. He was a labourer there for about six months until allowed to enter the University of Budapest in October 1940 to study mathematics and physics. His undergraduate lecturers included Lipót Fejér and Paul Turán who writes [19]:-

I first met him in October or November of 1941 when he ... started regularly to attend a lecture series I held to June of 1942. The main topic of discussion was number theory ...

In May 1944 Rényi graduated from Budapest University but, in the following month, he was forced into a Fascist Labour Camp but somehow managed to escape [19]:-

Fortunately, his company was not taken from Hungary immediately. When the order to evacuate to the West did come, Rényi escaped, and lived in Budapest using false documents.

He obtained false papers and hid for six months managing to avoid capture. During this time his parents were held prisoners in the Budapest ghetto. Alfréd rescued them with an extreme act of bravery [20]:-

Alfréd got hold of a soldier's uniform, walked into the ghetto, and marched his parents out. ... It requires familiarity with the circumstances to appreciate the skill and courage needed to perform these feats.

Turán writes [19]:-

Whenever I met him during those days, I was amazed at his level-headedness and courage.

Near the end of World War II in March 1945, Rényi obtained a Ph.D. from the University of Szeged, with Frigyes Riesz as his thesis advisor, for a thesis on Cauchy-Fourier series. Results from his doctoral thesis appeared in the paper On the summability of Cauchy-Fourier series (1950). He left Szeged after the award of his doctorate and returned to Budapest where he took on various statistics jobs. During his time in Budapest, he married Katalin Schulhof (1924-1969) in 1946. They had one child, a daughter Zsuzsa born in 1948. Katalin (known also as Catherine or Kató) was also a mathematician and wrote 21 papers, some with her husband. As well as undertaking research in analysis she was very successful in encouraging young university students in their research in all branches of mathematics. The János Bolyai Mathematical Society awards the Kató Rényi Memorial Prize in her honour.

Rényi went to Russia as a postdoctoral student and, between October 1946 and June 1947, worked with Yuri Vladimirovich Linnik on the theory of numbers, in particular working on the Goldbach conjecture [19]:-

His development during those months was nothing short of phenomenal. By an effort of the will, he had effaced his memories of the war years and of the forced-labour camp, to centre now on his work all the fiery energy of his youth and of his exceptional gifts of understanding and concentration.

He discovered methods described by Turán as:-

... at present one of the strongest methods of analytical number theory.

The results Rényi obtained in Russia were announced in the paper On the representation of an even number as the sum of a single prime and a single almost-prime number (Russian) (1947). L Schoenfeld summarises these results as follows:-

A set of integers $S$, with the property that there exists an absolute constant $K$ such that each $x$ in $S$ has at most $K$ distinct prime factors, is called an almost-prime set. Each $x$ in $S$ is called an almost-prime number. The author indicates the proof, to be given in detail elsewhere, that each even integer is the sum of an almost-prime number (taken from a fixed set $S$) and a prime number. He also states that he can prove that there exist infinitely many primes $p$ such that $p + 2$ is almost-prime (being in a fixed set $S^{*}$). The first result, regarding the representation of an even number, is an approximation to the unproved Goldbach conjecture and supersedes an earlier proof of the same proposition by Estermann (1932) which made use of an unproved generalized Riemann hypothesis for all Dirichlet $L$-series. The second result is an approximation to the conjecture of the existence of infinitely many twin primes and is apparently a new result.

In the following year Rényi published full proofs of these results and, in addition, noted that his techniques also show that every odd number is the sum of a prime and twice an almost-prime and that for each fixed integer m (positive or negative) there exists an infinity of primes $p$ such that $p + m$ is almost prime.

Other papers published early in his career include: On a Tauberian theorem of O Szász (1946); Integral formulae in the theory of convex curves (1947); On the minimal number of terms of the square of a polynomial (1947); On some new applications of the method of Academician I M Vinogradov (1947); (with Yu V Linnik) On certain hypotheses in the theory of Dirichlet characters (Russian) (1947). After returning to Hungary he was appointed as a Privatdocent and Assistant Professor in October 1947 at the University of Budapest. Between 1949 and October 1950 he was an Extraordinary Professor at the university in Debrecen. He was elected to the Hungarian Academy of Sciences in 1949 as a corresponding member, In 1950 he was appointed as the Director of the new Institute of Applied Mathematics of the Hungarian Academy of Sciences. In 1952, in addition to his other roles, he was appointed as a professor at the Department of Probability and Statistics of Eötvös Loránd University in Budapest.

Rényi worked on probability theory which was to be his main research topic throughout his life, but his interests were broad and also covered statistics, information theory, combinatorics, graph theory, number theory and analysis. His list of publications contains, remarkably, 355 items. David Kendall writes [1]:-

In the hands of writers like Linnik, Erdős and Rényi, the theory of numbers is not clearly distinguished from the theory of probability. Each lends techniques to the other, and important problems lie along their common frontier. Thus, when Rényi is referred to as a great applied probabilist, this is partly because of his interests in probability applied to other parts of mathematics.

He published joint work with Erdős on random graphs, the most important being On the evolution of random graphs (1960), and also solved an outstanding conjecture concerning random space filling curves in On a one-dimensional problem concerning random space filling (1958). He also produced a number of outstanding books including The calculus of probabilities (Hungarian) (1954). Eugene Lukacs writes:-

Probability theory and its applications had been neglected in the curriculum of Hungarian universities until very recently when the author started to lecture regularly on these topics. He had therefore to prepare mimeographed notes for his students which lead, after repeated revisions, to the publication of the present book. It is thus the first modern Hungarian text book on probability theory and offers an excellent introduction into this field. ... It is regrettable that this book is useful only to Hungarian students; it would deserve to be added to the foreign language publications of the Hungarian Academy of Sciences.

Indeed in 1962 a thoroughly revised and completely reorganized German edition was published under the title Wahrscheinlichkeitsrechnung. Mit einem Anhang über Informationstheorie. A French edition appeared in 1966 and an English edition, containing three new sections, was published as Probability theory in 1970. Another book, also published in 1970, was Foundations of probability. This book developed a totally different approach to probability, based on the concept of conditional probability space, from any other book previously published on the topic. This approach was based on a new system of axioms which Rényi had invented and presented in a lecture to the International Congress of Mathematicians held in Amsterdam from 2 September to 9 September 1954.

Known to his many friends and colleagues by the nickname of 'Buba,' he often remembered as the author of the anecdote:-

... a mathematician is a machine for converting coffee into theorems

Turán developed the anecdote further by describing weak coffee as fit only for lemmas. Rényi was a famous raconteur remembered for many performances of his dialogue, which he addressed to his daughter, on the nature of mathematics. In this style he published Dialoge über Mathematik Ⓣ (1967) and Letters on probability (Hungarian edition 1969, English translation 1972). This work is a fascinating semi-popular, semi-historical account of some of the early ideas on probability. Rényi adopts the style of presentation as a collection of (fictitious) letters from Pascal to Fermat. He assumes that the replies have been lost. To give a flavour of Dialoge über Mathematik Ⓣ we quote what Rényi has Archimedes say to King Hieron [2]:-

Mathematics is like your daughter Helena, who suspects every time a suitor appears that he is not really in love with her, but is interested in her only because he wants to be the son-in-law of the king. She wants a husband who loves her for her own beauty, her wit and charm, and not for the wealth and power he can get by marrying her. Similarly, mathematics reveals its secrets only to those who approach it with pure love, for its own beauty. Those who do this are, of course, also rewarded with results of practical importance. But if somebody asks at each step "What can I profit by this?" he will not get far.

Rényi received many honours for his achievements and, had he not died at the tragically young age of 48, he would have undoubtedly have received many more. He was twice awarded the Kossuth Prize by the Hungarian Government, was elected vice-president of the International Statistical Institute, served as he was secretary of the János Bolyai Mathematical Society (1949-55), and was invited to serve on the editorial boards of eight journals. After his sudden death, material was found for a book on which he was working Diary on information theory. Gyula Katona, using Rényi's notes for the rest of the book, completed it and it was published first in Hungarian, then in German in 1982, and in English in 1984, fourteen years after Rényi's death.

David Kendall in [10] gives us a little more understanding into Rényi's ways of working. He was:-

... a pure mathematician of massive achievements and towering stature in the classical fields of number theory and analysis. Rényi possessed also an inquisitive and dogged interest in all the phenomena of the world about him, and in all the scholarly activities of his colleagues, whether scientific or humane, and this unique combination of powers and interests enabled him to build up a research institute in which the criterion for acceptability of a subject for investigation was 'does there exits at least one mathematician with a genuine interest in this topic'? Once accepted as appropriate, however, the topic would be pursued in a thoroughly professional way; the argument would be followed wherever it led, and buttressed by whatever mathematical means seemed appropriate, however exotic or sophisticated.

Finally let us quote Turán concerning Rényi's leisure interests [19]:-

Rényi's zest for life was by no means exhausted in his many-sided intellectual activities and involvement in public affairs. He was fond of rowing and swimming in the Danube in summer and of skiing in winter. With his wife and daughter Zsuzsa he frequented concerts and theatres; at the parties they gave in their home, Rényi entertained his friends with witty anecdotes and with playing the piano. Those entering his study were welcomed by bookshelves jammed to the ceiling. The books, manuscripts and notes scattered on his desk made the visitor feel that he had entered the scene of creative, productive activity, an activity that Alfréd Rényi carried on unabated to the last day of his life.