约翰·彼得·古斯塔夫·勒热纳·狄利克雷Johann Peter Gustav Lejeune Dirichlet

约翰·彼得·古斯塔夫·勒热纳·狄利克雷的家族来自比利时小镇里什莱，狄利克雷的祖父就住在那里。这解释了他名字的起源，它来自“Le jeune de Richelet”，意思是“来自里什莱的年轻人”。狄利克雷家族的许多细节在[6]中给出，其中表明狄利克雷家族来自比利时列日附近，而不是像许多人声称的那样来自法国。

他的父亲是迪伦的邮政局长，他出生在这个位于亚琛和科隆之间大约中点的小镇。甚至在1817年他12岁进入波恩的文理中学之前，他就已经对数学产生了强烈的热情，并把零花钱花在购买数学书上。在文理中学，他是一个模范学生，[1]：-

……一个异常专注且品行端正的学生，对历史和数学都特别感兴趣。

在波恩的文理中学读了两年后，他的父母决定宁愿让他去科隆的耶稣会学院就读，在那里他有幸受教于格奥尔格·欧姆。到16岁时，狄利克雷已经完成了他的学校资格，准备进入大学。然而，当时德国大学的标准并不高，所以狄利克雷决定去巴黎学习。有趣的是，若干年后德国大学的标准将成为世界上最好的，而狄利克雷本人也将在这一转变中发挥作用。

狄利克雷带着卡尔·弗里德里希·高斯的Disquisitiones arithmeticaeⓉ（算术研究）出发前往法国，这是一部他珍视并像别人对待圣经一样随身携带的著作。1822年5月到达巴黎后，狄利克雷很快染上了天花。这并没有让他长时间远离法兰西学院和理学院的讲座，很快他就能回到讲座中。他有一些顶尖的数学家作为老师，并且能够从与让-巴蒂斯特·毕奥、约瑟夫·傅里叶、Francoeur、Hachette、皮埃尔·西蒙·拉普拉斯、西尔维斯特·佛朗索瓦·拉克鲁瓦、阿德里安-马里·勒让德和西莫恩·德尼·泊松接触的经历中获益良多。

从1823年夏天起，狄利克雷受雇于马克西米利安·塞巴斯蒂安·福伊将军，住在他在巴黎的家中。福伊将军在拿破仑战争期间是军队中的重要人物，在拿破仑滑铁卢战败后退役。1819年，他当选为众议院议员，在那里他一直是自由反对派的领袖，直到去世。狄利克雷受到福伊将军的很好对待，他薪水优厚，却被当作家庭成员一样对待。作为回报，狄利克雷教福伊将军的妻子和孩子们德语。

狄利克雷的第一篇论文使他立即成名，因为它涉及著名的费马大定理。该定理声称对于$n > 2$，不存在非零整数$x, y, z$使得$x^{n} + y^{n} = z^{n}$。情况$n = 3$和$n = 4$已由莱昂哈德·欧拉和皮埃尔·德·费马证明，而狄利克雷针对$n = 5$攻击了该定理。如果$n = 5$，那么$x, y, z$中有一个是偶数，一个能被5整除。有两种情况：情况1是能被5整除的数是偶数，而情况2是偶数和能被5整除的数不同。狄利克雷证明了情况1，并于1825年7月向巴黎科学院提交了他的论文。阿德里安-马里·勒让德被任命为裁判之一，他能够证明情况2，从而完成了$n = 5$的证明。完整的证明于1825年9月发表。事实上，狄利克雷能够完成他自己对$n = 5$情况的证明，其情况2的论证是他自己情况1论证的扩展。值得注意的是，狄利克雷后来做出了贡献，证明了$n = 14$情况（与$n = 7$情况擦肩而过！）。

1825年11月28日，福伊将军去世，狄利克雷决定返回德国。亚历山大·冯·洪堡鼓励他这样做，并为他做了推荐。狄利克雷面临一个问题，因为要在德国大学任教，他需要一份教授资格论文（Habilitation）。虽然狄利克雷可以轻松提交一篇任教资格论文，但这是不允许的，因为他没有博士学位，也不会说拉丁语，这是19世纪初的一项要求。这个问题被科隆大学很好地解决了，他们授予狄利克雷荣誉博士学位，从而允许他向布雷斯劳大学提交关于具有特殊类prime除子的多项式的任教资格论文。然而，关于狄利克雷的任命存在很多争议，德国教授之间支持和反对他任命的大量通信在[15]中进行了讨论。

从1827年起，狄利克雷在布雷斯劳任教，但狄利克雷遇到了使他选择巴黎接受自己教育的同样问题，即大学的标准很低。再次在冯·洪堡的帮助下，他于1828年搬到柏林，在那里他被任命在军事学院。当然，吸引他的不是军事学院，而是狄利克雷有一项协议，他将能够在柏林大学任教。此后不久，他被任命为柏林大学的教授，从1828年到1855年一直留在那里。他保留了在军事学院的职位，这使他的教学和其他行政职责比他希望的要繁重得多。

狄利克雷于1831年被任命到柏林科学院，大学提供的不断改善的薪水使他得以结婚，他娶了作曲家费利克斯·门德尔松的两个姐妹之一丽贝卡·门德尔松。狄利克雷与在柯尼斯堡任教的卡尔·古斯塔夫·雅各布·雅可比是终身好友，两人在数论研究中相互产生了相当大的影响。

1843年，卡尔·古斯塔夫·雅各布·雅可比身体不适，被诊断出患有糖尿病。他的医生建议他去意大利，那里的气候有助于他康复。然而，卡尔·古斯塔夫·雅各布·雅可比并不富有，狄利克雷在拜访卡尔·古斯塔夫·雅各布·雅可比并发现他的困境后，写信给亚历山大·冯·洪堡，请求他帮助从弗里德里希·威廉四世那里为卡尔·古斯塔夫·雅各布·雅可比争取一些经济援助。狄利克雷随后向弗里德里希·威廉四世提出援助请求，得到了亚历山大·冯·洪堡的大力支持，并获得了成功。狄利克雷从柏林获得十八个月的休假，并于1843年秋天与卡尔·古斯塔夫·雅各布·雅可比和卡尔·威廉·博尔夏特一起出发前往意大利。在几个城镇停留并参加了在卢卡举行的一次数学会议后，他们于1843年11月16日抵达罗马。路德维希·施莱夫利和雅各布·施泰纳也与他们同行，路德维希·施莱夫利的主要任务是担任他们的翻译，但他以狄利克雷为导师学习数学。

狄利克雷并没有在罗马待满整个时期，而是访问了西西里，然后在佛罗伦萨度过了1844/45年的冬天，之后于1845年春天返回柏林。狄利克雷在柏林大学有很高的教学负担，还被要求在军事学院授课，1853年他在给其学生利奥波德·克罗内克的信中抱怨说，除了许多其他职责外，他每周还要讲十三次课。因此，当1855年卡尔·弗里德里希·高斯去世后，他获得哥廷根大学的讲席职位时，这多少是一种解脱。

狄利克雷没有立即接受哥廷根的邀请，而是利用它试图在柏林获得更好的条件。他向普鲁士文化部请求允许他结束在军事学院的授课。然而，他这一 modest 请求没有得到迅速答复，于是他写信给哥廷根，接受了卡尔·弗里德里希·高斯讲席的邀请。在他接受哥廷根的邀请后，普鲁士文化部确实试图向他提供改善的条件和薪水，但这来得太晚了。

哥廷根较为安静的生活似乎适合狄利克雷。他有更多时间进行研究，并有一些杰出的研究学生。然而，遗憾的是，他未能长久享受这种新生活。1858年夏天，他在蒙特勒的一次会议上讲学，但在瑞士小镇期间，他心脏病发作。他极其艰难地返回哥廷根，在重病期间，更添悲伤的是他的妻子因中风去世。

我们现在应该看看狄利克雷对数学的杰出贡献。我们已经评论过他在1825年对皮埃尔·德·费马最后定理的贡献。大约在这个时候，他还发表了一篇受卡尔·弗里德里希·高斯关于双二次互反律工作启发的论文。细节见[13]，其中Rowe讨论了卡尔·弗里德里希·高斯与狄利克雷之间智识和个人关系的重要性。

他在1837年证明了在任何首项为互素到差值的等差数列中有无穷多个素数。这曾由阿德里安-马里·勒让德猜想。哈罗德·达文波特在1967年写道（见[16]）：-

可以说解析数论始于狄利克雷的工作，特别是始于狄利克雷1837年关于给定等差数列中素数存在的论文。

发表这篇论文后不久，狄利克雷又发表了两篇关于解析数论的论文，一篇在1838年，下一篇在次年。这些论文引入了狄利克雷级数，并除其他外，确定了二次型的类数公式。

他关于代数数论Vorlesungen über ZahlentheorieⓉ（数论讲义）（1863年出版）中单位的工作包含关于理想的重要工作。他还在1837年提出了函数的现代定义：-

如果一个变量y与一个变量x如此相关，以至于每当给x赋予一个数值时，就有一条规则根据它确定y的唯一值，那么y被称为自变量x的函数。

在力学中，他研究了系统的平衡以及potential theory。这些研究始于1839年的论文，其中给出了计算多重积分的方法，并将其应用于椭球对内外点的引力问题。他转向了皮埃尔·西蒙·拉普拉斯关于证明太阳系稳定性的问题，并给出了一种分析，避免了使用忽略二次及更高阶项的级数展开。这项工作引导他研究关于给定boundary conditions的调和函数的狄利克雷问题。他职业生涯后期的一些力学工作具有极其突出的重要性。1852年，他研究了置于不可压缩流体中的球体问题，在此研究过程中成为第一个精确积分流体动力学方程的人。

狄利克雷也以其关于三角级数收敛条件以及使用级数表示任意函数的论文而闻名。这些级数此前曾被约瑟夫·傅里叶用于求解微分方程。狄利克雷的工作于1828年发表在Crelle's Journal上。西莫恩·德尼·泊松早期关于Fourier series收敛性的工作被奥古斯丁·路易·柯西证明是不严谨的。奥古斯丁·路易·柯西的工作本身被狄利克雷指出有误，后者在写到奥古斯丁·路易·柯西的论文时说：-

这部作品的作者本人承认，对于某些函数，他的证明是有缺陷的，然而这些函数的收敛性却是无可争议的。

由于这项工作，狄利克雷被认为是约瑟夫·傅里叶级数理论的创始人。波恩哈德·黎曼是狄利克雷的学生，他在关于约瑟夫·傅里叶级数的任教资格论文的引言中写道，是狄利克雷[11]：-

……他撰写了关于该主题的第一篇深刻论文。

在[1]中，狄利克雷的性格和教学品质被总结如下：-

他是一位出色的教师，总是表达得极为清晰。他举止谦逊；晚年时他害羞，有时显得拘谨。他很少在会议上发言，也不愿公开露面。

45岁时，狄利克雷被托马斯·阿彻·赫斯特描述如下：-

他身材较高，瘦长，胡须和络腮胡即将变灰，声音有些刺耳，相当耳背。他不修边幅，带着咖啡杯和雪茄。他的缺点之一是忘记时间，他掏出怀表，发现已过三点，连话都没说完就跑了出去。

海里格·冯·科赫在[11]中总结狄利克雷的贡献时写道：-

……数学的重要部分受到了狄利克雷的影响。他的证明典型地始于令人惊讶的简单观察，随后对剩余问题进行极其敏锐的分析。随着狄利克雷，柏林数学的黄金时代开始了。

Lejeune Dirichlet's family came from the Belgium town of Richelet where Dirichlet's grandfather lived. This explains the origin of his name which comes from "Le jeune de Richelet" meaning "Young from Richelet". Many details of the Dirichlet family are given in [6] where it is shown that the Dirichlets came from the neighbourhood of Liège in Belgium and not, as many had claimed, from France.

His father was the postmaster of Düren, the town of his birth situated about halfway between Aachen and Cologne. Even before he entered the Gymnasium in Bonn in 1817, at the age of 12, he had developed a passion for mathematics and spent his pocket-money on buying mathematics books. At the Gymnasium he was a model pupil being [1]:-

... an unusually attentive and well-behaved pupil who was particularly interested in history as well as mathematics.

After two years at the Gymnasium in Bonn his parents decided that they would rather have him attend the Jesuit College in Cologne and there he had the good fortune to be taught by Ohm. By the age of 16 Dirichlet had completed his school qualifications and was ready to enter university. However, the standards in German universities were not high at this time so Dirichlet decided to study in Paris. It is interesting to note that some years later the standards in German universities would become the best in the world and Dirichlet himself would play a hand in the transformation.

Dirichlet set off for France carrying with him Gauss's Disquisitiones arithmeticae Ⓣ a work he treasured and kept constantly with him as others might do with the Bible. In Paris by May 1822, Dirichlet soon contracted smallpox. It did not keep him away from his lectures in the Collège de France and the Faculté des Sciences for long and soon he could return to lectures. He had some of the leading mathematicians as teachers and he was able to profit greatly from the experience of coming in contact with Biot, Fourier, Francoeur, Hachette, Laplace, Lacroix, Legendre, and Poisson.

From the summer of 1823 Dirichlet was employed by General Maximilien Sébastien Foy, living in his house in Paris. General Foy had been a major figure in the army during the Napoleonic Wars, retiring after Napoleon's defeat at Waterloo. In 1819 he was elected to the Chamber of Deputies where he was leader of the liberal opposition until his death. Dirichlet was very well treated by General Foy, he was well paid yet treated like a member of the family. In return Dirichlet taught German to General Foy's wife and children.

Dirichlet's first paper was to bring him instant fame since it concerned the famous Fermat's Last Theorem. The theorem claimed that for $n > 2$ there are no non-zero integers $x, y, z$ such that $x^{n} + y^{n} = z^{n}$. The cases $n = 3$ and $n = 4$ had been proved by Euler and Fermat, and Dirichlet attacked the theorem for $n = 5$. If $n = 5$ then one of $x, y, z$ is even and one is divisible by 5. There are two cases: case 1 is when the number divisible by 5 is even, while case 2 is when the even number and the one divisible by 5 are distinct. Dirichlet proved case 1 and presented his paper to the Paris Academy in July 1825. Legendre was appointed one of the referees and he was able to prove case 2 thus completing the proof for $n = 5$. The complete proof was published in September 1825. In fact Dirichlet was able to complete his own proof of the $n = 5$ case with an argument for case 2 which was an extension of his own argument for case 1. It is worth noting that Dirichlet made a later contribution proving the $n = 14$ case (a near miss for the $n = 7$ case!).

On 28 November 1825 General Foy died and Dirichlet decided to return to Germany. He was encouraged in this by Alexander von Humboldt who made recommendations on his behalf. There was a problem for Dirichlet since in order to teach in a German university he needed an habilitation. Although Dirichlet could easily submit an habilitation thesis, this was not allowed since he did not hold a doctorate, nor could he speak Latin, a requirement in the early nineteenth century. The problem was nicely solved by the University of Cologne giving Dirichlet an honorary doctorate, thus allowing him to submit his habilitation thesis on polynomials with a special class of prime divisors to the University of Breslau. There was, however, much controversy over Dirichlet's appointment and the large correspondence between German professors both for and against his appointment is considered in [15].

From 1827 Dirichlet taught at Breslau but Dirichlet encountered the same problem which made him choose Paris for his own education, namely that the standards at the university were low. Again with von Humboldt's help, he moved to the Berlin in 1828 where he was appointed at the Military College. The Military College was not the attraction, of course, rather it was that Dirichlet had an agreement that he would be able to teach at the University of Berlin. Soon after this he was appointed a professor at the University of Berlin where he remained from 1828 to 1855. He retained his position in the Military College which made his teaching and other administrative duties rather heavier than he would have liked.

Dirichlet was appointed to the Berlin Academy in 1831 and an improving salary from the university put him in a position to marry, and he married Rebecca Mendelssohn, one of the composer Felix Mendelssohn's two sisters. Dirichlet had a lifelong friend in Jacobi, who taught at Königsberg, and the two exerted considerable influence on each other in their researches in number theory.

In the 1843 Jacobi became unwell and diabetes was diagnosed. He was advised by his doctor to spend time in Italy where the climate would help him recover. However, Jacobi was not a wealthy man and Dirichlet, after visiting Jacobi and discovering his plight, wrote to Alexander von Humboldt asking him to help obtain some financial assistance for Jacobi from Friedrich Wilhelm IV. Dirichlet then made a request for assistance from Friedrich Wilhelm IV, supported strongly by Alexander von Humboldt, which was successful. Dirichlet obtained leave of absence from Berlin for eighteen months and in the autumn of 1843 set off for Italy with Jacobi and Borchardt. After stopping in several towns and attending a mathematical meeting in Lucca, they arrived in Rome on 16 November 1843. Schläfli and Steiner were also with them, Schläfli's main task being to act as their interpreter but he studied mathematics with Dirichlet as his tutor.

Dirichlet did not remain in Rome for the whole period, but visited Sicily and then spent the winter of 1844/45 in Florence before returning to Berlin in the spring of 1845. Dirichlet had a high teaching load at the University of Berlin, being also required to teach in the Military College and in 1853 he complained in a letter to his pupil Kronecker that he had thirteen lectures a week to give in addition to many other duties. It was therefore something of a relief when, on Gauss's death in 1855, he was offered his chair at Göttingen.

Dirichlet did not accept the offer from Göttingen immediately but used it to try to obtain better conditions in Berlin. He requested of the Prussian Ministry of Culture that he be allowed to end lecturing at the Military College. However he received no quick reply to his modest request so he wrote to Göttingen accepting the offer of Gauss's chair. After he had accepted the Göttingen offer the Prussian Ministry of Culture did try to offer him improved conditions and salary but this came too late.

The quieter life in Göttingen seemed to suit Dirichlet. He had more time for research and some outstanding research students. However, sadly he was not to enjoy the new life for long. In the summer of 1858 he lectured at a conference in Montreux but while in the Swiss town he suffered a heart attack. He returned to Göttingen, with the greatest difficulty, and while gravely ill had the added sadness that his wife died of a stroke.

We should now look at Dirichlet's remarkable contributions to mathematics. We have already commented on his contributions to Fermat's Last Theorem made in 1825. Around this time he also published a paper inspired by Gauss's work on the law of biquadratic reciprocity. Details are given in [13] where Rowe discusses the importance of the intellectual and personal relationship between Gauss and Dirichlet.

He proved in 1837 that in any arithmetic progression with first term coprime to the difference there are infinitely many primes. This had been conjectured by Legendre. Davenport wrote in 1967 (see [16]):-

Analytic number theory may be said to begin with the work of Dirichlet, and in particular with Dirichlet's memoir of 1837 on the existence of primes in a given arithmetic progression.

Shortly after publishing this paper Dirichlet published two further papers on analytic number theory, one in 1838 with the next in the following year. These papers introduce Dirichlet series and determine, among other things, the formula for the class number for quadratic forms.

His work on units in algebraic number theory Vorlesungen über Zahlentheorie Ⓣ (published 1863) contains important work on ideals. He also proposed in 1837 the modern definition of a function:-

If a variable y is so related to a variable x that whenever a numerical value is assigned to x, there is a rule according to which a unique value of y is determined, then y is said to be a function of the independent variable x.

In mechanics he investigated the equilibrium of systems and potential theory. These investigations began in 1839 with papers which gave methods to evaluate multiple integrals and he applied this to the problem of the gravitational attraction of an ellipsoid on points both inside and outside. He turned to Laplace's problem of proving the stability of the solar system and produced an analysis which avoided the problem of using series expansion with quadratic and higher terms disregarded. This work led him to the Dirichlet problem concerning harmonic functions with given boundary conditions. Some work on mechanics later in his career is of quite outstanding importance. In 1852 he studied the problem of a sphere placed in an incompressible fluid, in the course of this investigation becoming the first person to integrate the hydrodynamic equations exactly.

Dirichlet is also well known for his papers on conditions for the convergence of trigonometric series and the use of the series to represent arbitrary functions. These series had been used previously by Fourier in solving differential equations. Dirichlet's work is published in Crelle's Journal in 1828. Earlier work by Poisson on the convergence of Fourier series was shown to be non-rigorous by Cauchy. Cauchy's work itself was shown to be in error by Dirichlet who wrote of Cauchy's paper:-

The author of this work himself admits that his proof is defective for certain functions for which the convergence is, however, incontestable.

Because of this work Dirichlet is considered the founder of the theory of Fourier series. Riemann, who was a student of Dirichlet, wrote in the introduction to his habilitation thesis on Fourier series that it was Dirichlet [11]:-

... who wrote the first profound paper about the subject.

In [1] Dirichlet's character and teaching qualities are summed up as follows:-

He was an excellent teacher, always expressing himself with great clarity. His manner was modest; in his later years he was shy and at times reserved. He seldom spoke at meetings and was reluctant to make public appearances.

At age 45 Dirichlet was described by Thomas Hirst as follows:-

He is a rather tall, lanky-looking man, with moustache and beard about to turn grey with a somewhat harsh voice and rather deaf. He was unwashed, with his cup of coffee and cigar. One of his failings is forgetting time, he pulls his watch out, finds it past three, and runs out without even finishing the sentence.

Koch, in [11], sums up Dirichlet's contribution writing that:-

... important parts of mathematics were influenced by Dirichlet. His proofs characteristically started with surprisingly simple observations, followed by extremely sharp analysis of the remaining problem. With Dirichlet began the golden age of mathematics in Berlin.