利特尔伍德John Edensor Littlewood

利特尔伍德的父母是Edward Thornton 利特尔伍德和Sylvia Maud Ackland。极少数亲密的朋友在他年迈时叫他Jack，但除此之外人们会称呼他为利特尔伍德，这在当时的朋友之间并不罕见。与他有多年密切合作的戈弗雷·哈罗德·哈代会写信称他为“亲爱的L”。

他的父亲Edward Thornton 利特尔伍德也是一位数学家，1882年在剑桥大学数学荣誉学位考试中名列第九数学荣誉学位考试一等及格者（Wrangler），三年后他的长子弗瑞兹·约翰 利特尔伍德出生。Edward和Sylvia Littlewood后来有了三个儿子，第二个是Martin Wentworth Littlewood，后来学医，第三个儿子不幸在八岁时从桥上掉入湖中身亡。当约翰 利特尔伍德七岁时，他的父亲不得不在收到的两个职位中做出选择：一个是剑桥大学Magdalene学院的研究员职位，另一个是南非Wynberg一所新学校的校长职位。他选择了后者，1892年全家乘船前往南非。

利特尔伍德后来写道（例如见[7]）：-

我们在剑桥的生活会非常不同。……大学教师的孩子们养成了一种轻松的自信心……而事实上，我在群山、海洋和美丽的气候中度过了非常快乐的童年。

如果说气候和风景是一流的，那么年轻的利特尔伍德在南非所受的教育肯定不是。他的学校教师素质很差，他因所受的数学教学而困惑，以至于算术考试不及格。他仍然表现相当不错，年纪轻轻就进入了开普敦大学，但再次发现无法从并不出色的教学中受益。他的父亲很快意识到开普敦大学不会鼓励他儿子的数学才能开花结果，利特尔伍德的父母决定把儿子送回英国。我们应该提到，正如我们在下面更详细解释的那样，利特尔伍德一生中大部分时间都患有抑郁症，从他在学校时就开始了，这可能导致了他在南非教育体系中的艰难日子。

1900年，时年15岁的利特尔伍德回到英格兰，进入伦敦的圣保罗学校。他非常幸运，在那里遇到了一位杰出的数学教师Francis Macaulay[7]：-

这是一种带有大学氛围的教育；在任何学校，利特尔伍德都不可能获得更好的自立、知识和判断力的基础。他理解一致收敛，并能区分基本思想与操作技巧。

1902年12月，还在圣保罗学校时，利特尔伍德赢得了剑桥的奖学金。

1903年10月，利特尔伍德进入剑桥大学三一学院。他在[3]中写道：-

要想竞争高级牧马人，必须花三分之二的时间练习如何在限时内解决难题……我并不声称自己遭受过高尚的挫折。我随遇而安；我们玩的游戏对我来说很容易，我甚至在成功的技艺中感到一种满足。

他在三一学院的导师是Walter Rouse Ball，著名畅销书Mathematical Recreations and Essays的作者。在第一部分并列第一后，他于1906年完成第二部分，并在导师欧内斯特·巴恩斯的指导下开始研究。利特尔伍德迅速解决了欧内斯特·巴恩斯给他的第一个问题，随后欧内斯特·巴恩斯将黎曼假设作为他的下一个研究问题交给他。 Swinnerton-Dyer在利特尔伍德去世后于三一学院的纪念演讲中说：-

当时英国数学界的孤立与封闭程度令人惊讶，欧内斯特·巴恩斯竟然认为它适合哪怕最杰出的研究生，而利特尔伍德竟然毫不犹豫地着手研究它。

利特尔伍德从未后悔研究波恩哈德·黎曼假设，他说如果尝试一个太难的问题，那么最终总会证明出一些有趣的相关结果。

从1907年到1910年，他在曼彻斯特大学担任理查森讲师。1908年，他成为三一学院的研究员，并于同年获得史密斯奖，然后在1910年回到三一学院，填补阿尔弗雷德·诺思·怀特海基本上被迫离职后留下的职位。此后不久，肯定在1911年，但有人认为在1910年，利特尔伍德开始了与戈弗雷·哈罗德·哈代的著名合作，我们将在下面更详细地讨论。

利特尔伍德曾在皇家驻防炮兵服役。他的贡献极为重要，为了让他满意，还给予了一些特殊照顾，比如允许他与伦敦的朋友同住，以及穿制服时可以带伞！利特尔伍德本人在[14]中描述了这项战时工作。结果是，他提高了高射炮射程表的精度，并改进了在小仰角下求射程、飞行时间和弹道末端下降角的公式。爱德华·亚瑟·米尔恩描述了利特尔伍德如何能够发现一些技术，这些技术大大减少了进行这些精确导弹弹道计算所需的工作量。进行了试验以检验利特尔伍德的预测结果在实践中是否成立，爱德华·亚瑟·米尔恩写道：-

……令所有相关人员惊讶和欣喜的是，观测到的炮弹爆炸位置恰好落在利特尔伍德的弹道上，时间标记正确，观测误差极小。

利特尔伍德于1928年成为剑桥的Rouse Ball数学讲席教授。这个讲席是Walter Rouse Ball于1925年去世后通过遗赠设立的，利特尔伍德是其第一位担任者。这是一个特别合适的选择，不仅因为他杰出的数学贡献，还因为利特尔伍德在剑桥三一学院读本科时曾受教于Rouse Ball。作为Rouse Ball教授，利特尔伍德可以讲授自己选择的主题，不再需要参与常规教学。这是他喜欢的一个方面，他讲授自己在分析中广泛兴趣领域的课程。

利特尔伍德几乎所有的数学研究都在经典分析领域，但在这个领域中，他研究了极为广泛的主题，并且在证明结果时使用了更广泛的技术。35年来，他与戈弗雷·哈罗德·哈代合作研究级数理论、Riemann zeta function、不等式和函数论。这一合作产生了一系列使用戈弗雷·哈罗德·哈代-利特尔伍德-拉马努金分析方法的论文Partitio numerorum。在这些合作年间，利特尔伍德很少在剑桥以外露面，事实上周围有笑话称他是戈弗雷·哈罗德·哈代的发明。真正的原因相当令人悲伤，即利特尔伍德患有抑郁症，我们将在下面更全面地讨论。戈弗雷·哈罗德·哈代和利特尔伍德为他们的合作所采用的规则由哈拉尔·玻尔在1947年的一次演讲中阐明（例如见[11]，其中还有关于他们在多大程度上遵守自己的规则以及多大程度上忽视它们的有趣讨论）：-

1. When one wrote to the other, it was completely indifferent whether what they wrote was right or wrong。
2. When one received a letter from the other, he was under no obligation to read it, let alone answer it。
3. Although it did not really matter if they both simultaneously thought about the same detail, still it was preferable that they should not do so。
4. It was quite indifferent if one of them had not contributed the least bit to the contents of a paper under their common name。

在11中，玛丽·卡特赖特推测这些规则实际上是由利特尔伍德和戈弗雷·哈罗德·哈代在1912年商定的。我们还应当指出，如果像利特尔伍德这样多产的数学家，其论文集只出版了两卷，这看起来很奇怪，那是因为大量的戈弗雷·哈罗德·哈代-利特尔伍德论文集出现在戈弗雷·哈罗德·哈代的全集中。

20世纪30年代末，科学与工业研究部试图让纯数学家对非线性微分方程产生兴趣，因为非线性微分方程对无线电工程师和科学家很重要，它们描述了电路的行为。即将到来的战争促成了这种兴趣，1938年，无线电研究委员会请求英国纯数学家帮助处理无线电工程中出现的某些类型的非线性微分方程。利特尔伍德与玛丽·卡特赖特合作，花了20年时间研究这类方程，例如巴尔塔萨·范德波尔方程。McMurran和Tattersall在[16]中讨论了这一合作，特别是对巴尔塔萨·范德波尔方程的工作在[17]中进行了讨论。他们写道：-

巴尔塔萨·范德波尔在20世纪20年代和30年代对非线性振子的实验，激发了数学界对无线电研究中出现的非线性微分方程的兴趣。这些问题引起了英国数学家玛丽·卡特赖特和J 利特尔伍德的注意，开启了一项持续十多年的合作。玛丽·卡特赖特和利特尔伍德对巴尔塔萨·范德波尔方程及其推广的分析，使他们探索了一些有趣的拓扑方法，包括为平面同胚下不变的连续统发展不动点定理。他们是最早将儒勒·昂利·庞加莱变换理论应用于耗散系统分析的数学家之一。他们的研究属于大参数理论中最早的一批，并在现代动力系统和混沌理论的发展中发挥了作用。

利特尔伍德于1915年当选为皇家学会会士。他于1929年获得该学会的皇家奖章，并于1943年获得该学会的詹姆斯·约瑟夫·西尔维斯特奖章：-

……以表彰他的数学发现以及在解析数论中的卓越洞察力。……按照戈弗雷·哈罗德·哈代自己的评价，利特尔伍德是他所认识的最优秀的数学家。他最有可能攻克并粉碎一个真正深刻而艰巨的问题；没有其他人能拥有如此集洞察力、技巧和力量于一身的组合。

他还在1958年获得了该学会的科普利奖章：-

……以表彰他在分析的许多分支中作出的杰出贡献，包括陶伯尔理论、波恩哈德·黎曼 zeta函数和非线性微分方程。

正如我们在上面几次提到的那样，利特尔伍德一生中不得不应对的一个问题是抑郁症。它从学生时代起就影响着他，并在他的整个职业生涯中一直困扰着他。没有证据表明它以任何方式损害了他做数学的能力，事实上，鉴于他所取得的成就，很难看出它怎么可能做到这一点，但它确实在很大程度上破坏了他的社交生活，因为它使他变得内向。尽管他的数学研究似乎没有受到影响，但它确实阻止了他从事其他涉及与人会面的数学活动。例如，他在晚年能够说，他从未担任过任何委员会的秘书或主席。事实上，尽管他在1941-43年担任伦敦数学会主席，但他从未主持过会议，在他任职期间一直由副主席主持会议。

退休后，他确实接受了抑郁症治疗，效果相当不错，1957年后他觉得更能接受邀请去拜访同事。事实上，在1957年之后的十年里，他多次访问美国。我们注意到，他与人会面的问题并没有延伸到那些他熟悉的人，例如，他完全参与了三一学院的晚餐谈话，并在熟悉的环境中非常享受陪伴。

我们应该以关于利特尔伍德在数学之外的兴趣的一些内容来结束这本传记。他[7]：-

……身高略低于平均水平，体格强壮且敏捷。在学校时，他曾是最好的体操运动员之一，也是一名击球有力的板球击球手。……他热衷于球类运动，并在夏日午后到芬纳斯观看板球。他对音乐（古典音乐——尤其是巴赫、贝多芬和莫扎特）有着浓厚的兴趣；他在成年后自学弹钢琴……在三一学院的非数学本科生中，他最出名的是他能够沿着图书馆一根柱子底部狭窄的边缘绕行七码，以及他每天穿过庭院去浴池，只带一条毛巾而不穿上衣。大多数日子里，他会在乡间步行许多英里。

他的[2]：-

……肌肉力量和反应敏捷使他在攀岩和滑雪中取得成功，他在康沃尔、苏格兰和瑞士度过了许多假期。

退休后许多年他仍不断产出优秀的数学成果，甚至到了九十岁依然思维敏锐，能够提出新的深刻想法。1970年他发表了一篇论文，其中写道他解决了一个问题，该问题：-

……带来的困难曾使我一度受挫。我现在已经克服了它们。

除了我们上面提到的荣誉之外，利特尔伍德还于1925年当选哥廷根Akademie der Wissenschaften，1948年当选Swedish Academy，同年又当选丹麦皇家科学院，1950年当选荷兰科学院院士。1957年他当选巴黎Académie des Sciences，接替莫里斯·弗雷歇。我们上面提到利特尔伍德曾任伦敦数学会主席，但该学会还于1938年授予他奥古斯塔斯·德摩根奖章，并于1960年因他关于天体力学的两篇论文授予他高级贝里克奖。

最后我们注意到，他于1928年获利物浦大学荣誉学位，1936年获圣安德鲁斯大学荣誉学位，1965年获剑桥大学荣誉学位。

J E Littlewood's parents were Edward Thornton Littlewood and Sylvia Maud Ackland. A very few close friends called him Jack when he was elderly, but otherwise he would be addressed as Littlewood, which was not unusual between friends at that time. Hardy with whom he had a close collaboration for many years would write to him as "Dear L".

His father Edward Thornton Littlewood was also a mathematician and was Ninth Wrangler in the Mathematical Tripos at Cambridge in 1882, three years before his eldest son John Edensor was born. Edward and Sylvia Littlewood went on to have three sons, their second being Martin Wentworth Littlewood, who went on to study medicine, and a third son who tragically died when he was eight years old by falling into a lake from a bridge. When John Edensor was seven years old his father had to make a choice between two offers he received, one of a Fellowship at Magdalene College, Cambridge, or the second the position of Headmaster at a new school in Wynberg in South Africa. He chose the second and in 1892 the family sailed to South Africa.

Littlewood later wrote (see for example [7]):-

Our lives would have been very different in Cambridge. ... the children of dons acquire an easy self-confidence ... As it was I had a very happy childhood among mountains, the ocean, and a beautiful climate.

If the climate and scenery were superb, certainly the education that the young Littlewood had in South Africa was not. The quality of his school teachers was poor, and he was confused by the mathematics teaching that he received to the extent that he failed an arithmetic examination. He still did quite well and entered the University of Cape Town at a young age but again found that he was not able to benefit from teaching which was less than outstanding. His father soon realised that the University of Cape Town was not going to encourage his son's mathematical talents to blossom and Littlewood's parents made the decision to send their son back to England. We should mention that Littlewood, as we explain in more detail below, suffered from depression for most of his life, beginning while he was at school, and this may have contributed to his difficult days in the South African education system.

In 1900 Littlewood, then aged 15, returned to England and entered St Paul's School in London. He was very fortunate to have there an outstanding teacher of mathematics Francis Macaulay [7]:-

This was education with a university atmosphere; at no school could Littlewood have acquired a better foundation of self-reliance, knowledge and judgement. He understood uniform convergence, and he could discriminate between basic ideas and tricks of manipulation.

While at St Paul's School in December 1902, Littlewood won a scholarship to Cambridge.

Littlewood entered Trinity College Cambridge in October 1903. He wrote in [3]:-

To be in the running for Senior Wrangler one had to spend two-thirds of the time practising how to solve difficult problems against time ... I do not claim to have suffered high-souled frustration. I took things as they came; the game we were playing came easily to me, and I even felt a sort of satisfaction in successful craftsmanship.

His tutor at Trinity was Walter Rouse Ball, the author of the famous popular book Mathematical Recreations and Essays. After being equal first in Part I, he completed Part II in 1906 and began research under the direction of his tutor E W Barnes. Rapidly solving the first problem which Barnes gave him, Littlewood was next presented with the Riemann hypothesis as his next research problem by Barnes. Swinnerton-Dyer, in a memorial address at Trinity College after Littlewood's death, said:-

It is an amazing illustration of the isolation and insularity of British mathematics at that time that Barnes should have thought it suitable for even the most brilliant research student, and that Littlewood should have tackled it without demur.

Littlewood never regretted having tackled the Riemann hypothesis, remarking that if one attempted a problem that was too difficult then one would always end up proving some interesting related results.

From 1907 to 1910 he lectured as Richardson Lecturer at the University of Manchester. He became a fellow of the Trinity College in 1908, winning a Smith's prize in that year, then returning to Trinity in 1910 to fill the position left vacant when A N Whitehead was essentially forced out of his job. Shortly after this, certainly by 1911 but some suggest in 1910, Littlewood began his famous collaboration with G H Hardy which we discuss in more detail below.

In World War I Littlewood served in the Royal Garrison Artillery. His contributions were highly significant and special allowances were made to keep him happy, such as letting him live with friends in London, and to carry an umbrella when in uniform! Littlewood himself described this war work in [14]. The result was that he improved the accuracy of anti-aircraft range tables and improved the formulae for finding the range, the time of flight and the angle of descent at the end of a trajectory with small elevation. E A Milne has described how Littlewood was able to discover techniques which greatly reduced the amount of work needed for making these accurate calculation of missile trajectories. Trials were conducted to see if the results of Littlewood's predictions held in practice and, Milne writes:-

... to the astonishment and joy of all concerned the observed positions of the shellbursts fell exactly on Littlewood's trajectories, at the correct time-markings, within very small errors of observation.

Littlewood become Rouse Ball professor of mathematics in Cambridge in 1928. This chair had been founded by a benefaction from Walter Rouse Ball after his death in 1925, and Littlewood was its first occupant. It was a particularly appropriate choice, not only because of his outstanding mathematical contributions, but also since Littlewood had been tutored by Rouse Ball in his undergraduate days at Trinity. As Rouse Ball Professor, Littlewood could lecture on topics of his own choice and he no longer had to take part in routine teaching. It was an aspect which he enjoyed, delivering courses on his own wide areas of interest in analysis.

Almost all of Littlewood's mathematical research was in classical analysis, but in this area he looked at a remarkable range of subjects and he used an even broader range of techniques in proving his results. For 35 years he collaborated with G H Hardy working on the theory of series, the Riemann zeta function, inequalities, and the theory of functions. The collaboration led to a series of papers Partitio numerorum using the Hardy-Littlewood-Ramanujan analytical method. During the years of this collaboration Littlewood was seldom seen outside Cambridge, in fact there were jokes around that he was the invention of Hardy. The real reason was rather a sad one, namely that Littlewood suffered from depression which we shall discuss more fully below. The rules that Hardy and Littlewood adopted for their collaboration were spelled out by Harald Bohr in a lecture which he gave in 1947 (see for example [11] where there is also an interesting discussion as to how far they stuck to their own rules and how far they ignored them):-

1. When one wrote to the other, it was completely indifferent whether what they wrote was right or wrong.
2. When one received a letter from the other, he was under no obligation to read it, let alone answer it.
3. Although it did not really matter if they both simultaneously thought about the same detail, still it was preferable that they should not do so.
4. It was quite indifferent if one of them had not contributed the least bit to the contents of a paper under their common name.

In [11] Cartwright conjectures that these rules were actually agreed on by Littlewood and Hardy in 1912. We should also comment that if it seems strange that such a prolific mathematician as Littlewood has his collected papers published in only two volumes, this is because the large Hardy-Littlewood collection of papers appears in Hardy's collected works.

In the late 1930's the Department of Scientific and Industrial Research tried to interest pure mathematicians in nonlinear differential equations which were important for radio engineers and scientists because they described the behaviour of electric circuits. The impending war motivated this interest and in 1938 the Radio Research Board asked British pure mathematicians for help in dealing with certain types of nonlinear differential equations arising in radio engineering. Littlewood, working jointly with Mary Cartwright, spent 20 years working on equations of this type such as van der Pol's equation. McMurran and Tattersall discuss this collaboration in [16], and in particular the work on van der Pol's equation is discussed in [17]. They write:-

Van der Pol's experiments with nonlinear oscillators during the 1920s and 1930s stimulated mathematical interest in nonlinear differential equations arising in radio research. The problems caught the attention of British mathematicians M L Cartwright and J E Littlewood, initiating a collaboration that lasted more than ten years. Cartwright and Littlewood's analysis of the van der Pol equation and its generalizations led them to explore some interesting topological methods, including the development of a fixed-point theorem for continua invariant under a homeomorphism of the plane. They were among the earliest mathematicians to apply Poincare's transformation theory to the analysis of dissipative systems. Their research is among the earliest in large parameter theory and played a role in the development of the modern theory of dynamical systems and chaos theory.

Littlewood was elected a Fellow of the Royal Society in 1915. He received the Royal Medal of the Society in 1929, and the Sylvester Medal of the Society in 1943:-

... in recognition of his mathematical discoveries and supreme insight in the analytic theory of numbers. ... Littlewood, on Hardy's own estimate, is the finest mathematician he has ever known. He was the man most likely to storm and smash a really deep and formidable problem; there was no one else who could command such a combination of insight, technique and power.

He also received the Copley Medal of the Society in 1958:-

... in recognition of his distinguished contributions to many branches of analysis, including Tauberian theory, the Riemann zeta-function, and non-linear differential equations.

As we mentioned on a couple of occasions above, one problem which Littlewood had to cope with throughout his life was depression. It had affected him from his school days and continued to trouble him throughout his career. There is no evidence that it in any way impaired his ability to do mathematics, indeed it is hard to see how it could have done so given what he achieved, yet it certainly did much to spoil his social life for it had the effect of making him withdraw into himself. Although his mathematical research did not seem to suffer, it certainly stopped him from undertaking other mathematical activities which involved meeting people. For example he was able to say late in his life that he had never been secretary or chairman of any committee. In fact although he was President of the London Mathematical Society in 1941-43 he never chaired a meeting, the vice-President taking the chair throughout his period of office.

After he retired he did receive treatment for his depression which was quite effective, and after 1957 he felt more able to accept invitations to visit colleagues. Indeed he made many visits to the United States in the ten years following 1957. We note that his problems of meeting people did not extend to those whom he knew well, for example he fully participated in the conversations at dinner in Trinity and greatly enjoyed the company in the familiar surroundings.

We should end this biography by saying something about Littlewood's interests outside mathematics. He [7]:-

... was slightly below average in height, strongly built and agile. At school he had been one of the best gymnasts and a hard hitting batsman. ... he was a keen follower of ball games and watched cricket at Fenner's on summer afternoons. He had an intense interest in music (classical - particularly Bach, Beethoven and Mozart); he had taught himself as an adult to play the piano ... He was best known to unmathematical undergraduates at Trinity for his skill in circling the seven yards of a pillar of the Library on the narrow ledge of its base and for his daily walk across the court to the baths, with a towel but no shirt. On most days he walked many miles in the country.

His [2]:-

... muscular strength and quickness of reaction made for success in rock climbing and skiing, and he spent many holidays in Cornwall, Scotland and Switzerland.

He continued to produce excellent mathematical results for many years after he retired and even when ninety years old he was still sharp and able to come up with new deep ideas. In 1970 he published a paper in which he wrote that he had solved a problem which:-

... raised difficulties which defeated me for some time. I have now overcome them.

In addition to the honours which we have mentioned above, Littlewood was honoured with election to the Akademie der Wissenschaften in Göttingen in 1925, the Swedish Academy in 1948, the Royal Danish Academy also in 1948, and the Dutch Academy in 1950. In 1957 he was elected to the Paris Académie des Sciences to replace Fréchet. We mentioned above that Littlewood was President of the London Mathematical Society, but that Society also honoured him with its De Morgan Medal in 1938 and its Senior Berwick Prize in 1960 for two papers which he wrote on celestial mechanics.

Finally we note that he received honorary degrees from the University of Liverpool in 1928, the University of St Andrews in 1936 and the University of Cambridge in 1965.