拉马努金Srinivasa Ramanujan

拉马努金是印度最伟大的数学天才之一。他对数的分析理论做出了重大贡献，并研究了椭圆函数、连分数和无穷级数。

拉马努金出生在他祖母位于埃罗德的家中，这是马德拉斯（现金奈）西南约400公里的一个小村庄。当拉马努金一岁时，他的母亲带他到了昆巴科纳姆镇，那里离马德拉斯近了约160公里。他的父亲在昆巴科纳姆的一家布店当职员。1889年12月，他感染了天花。

当他快五岁时，拉马努金进入了昆巴科纳姆的小学，尽管在1898年1月进入昆巴科纳姆的镇立高中之前，他上过几所不同的小学。在镇立高中，拉马努金在所有学校科目上都表现出色，显示出他是一个能干的全面学者。1900年，他开始自学数学，对几何级数和算术级数求和。

1902年，有人向拉马努金展示了如何求解三次方程，他接着找到了自己的方法来求解quartic。第二年，在不知道quintic无法用根式求解的情况下，他尝试（当然失败了）求解五次方程。

正是在城镇中学，拉马努金偶然看到了一本G S Carr写的数学书，书名为Synopsis of elementary results in pure mathematics。这本书风格非常简洁，使拉马努金得以自学数学，但该书的风格对拉马努金后来书写数学的方式产生了相当不幸的影响，因为它为他提供了书面数学论证的唯一范本。书中包含定理、公式和简短证明，还附有一份19世纪上半叶发表在欧洲学术团体期刊上的纯数学论文索引。这本书出版于1886年，到拉马努金使用时自然早已过时。

到1904年，拉马努金已开始进行深入研究。他研究了级数$\sum (\Large\frac{1}{n}\normalsize )$，并将莱昂哈德·欧拉常数计算到15位小数。他开始研究伯努利数，尽管这完全是他独立发现的。

拉马努金凭借优异的学业成绩，获得了1904年入学的昆巴科纳姆政府学院的奖学金。然而，第二年他的奖学金未能续发，因为拉马努金越来越多地将时间投入数学，而忽视了其他科目。没有钱，他很快陷入困境，并且没有告诉父母，就逃到了马德拉斯以北约650公里的维沙卡帕特南镇。然而，他继续从事数学工作，此时他研究了hypergeometric series，并探讨了积分与级数之间的关系。他后来发现，自己一直在研究椭圆函数。

1906年，拉马努金前往马德拉斯，进入帕恰亚帕学院。他的目标是通过第一文科考试，以便能被马德拉斯大学录取。他在帕恰亚帕学院听课，但学习三个月后生病了。离开课程后，他参加了第一文科考试。他的数学及格，但其他所有科目都不及格，因此考试未通过。这意味着他无法进入马德拉斯大学。在接下来的几年里，他独自研究数学，发展自己的想法，没有任何帮助，除了Carr的书之外，对当时的研究课题也没有真正的了解。

继续从事数学工作，拉马努金在1908年研究了连分数和发散级数。在这个阶段，他再次重病，并于1909年4月接受手术，之后花了相当长的时间才康复。1909年7月14日，他结婚了，母亲安排他娶了一个十岁的女孩S Janaki Ammal。然而，拉马努金直到妻子十二岁时才与她同住。

拉马努金继续发展他的数学思想，并开始在Journal of the Indian Mathematical Society中提出问题并解决问题。1910年，他推导出了椭圆模方程之间的关系。1911年，在Journal of the Indian Mathematical Society上发表了一篇关于伯努利数的杰出研究论文后，他的工作得到了认可。尽管缺乏大学教育，他作为数学天才在马德拉斯地区已逐渐闻名。

1911年，拉马努金向印度数学会的创始人寻求工作建议。此后，他被任命为第一份工作，在马德拉斯总会计师办公室担任临时职位。然后有人建议他去找当时在内洛尔的税务官Ramachandra Rao。Ramachandra Rao是印度数学会的创始成员，曾帮助建立数学图书馆。他在[30]中写道：-

一个矮小粗陋的身影，胖胖的，没刮胡子，不太干净，有一个显眼的特征——闪亮的眼睛——腋下夹着一本磨损的笔记本走了进来。他穷困潦倒。……他打开书，开始解释他的一些发现。我立刻看出有些不同寻常；但我的知识不允许我判断他讲的是有道理还是胡说八道。……我问他想要什么。他说他想要一点微薄的收入来维持生活，以便继续他的研究。

Ramachandra Rao 让他回到马德拉斯，他尝试为 拉马努金 安排奖学金，但没有成功。1912年，拉马努金 申请了马德拉斯港务信托局会计科的文书职位。在他的申请信中，他写道[3]：-

我已通过入学考试并学习到文科一年级，但由于一些不幸的情况，未能继续深造。然而，我一直将所有时间投入到数学中，并发展这一学科。

尽管他没有接受过大学教育，拉马努金 显然为马德拉斯的大学数学家们所熟知，因为在他的申请信中，拉马努金 附上了马德拉斯管区学院数学教授 E W Middlemast 的推荐信。Middlemast 是剑桥圣 弗瑞兹·约翰 学院的毕业生，他写道[3]：-

我强烈推荐这位申请人。他在数学方面，尤其是在与数字相关的工作中，具有非凡的能力。他天生擅长计算，对数字工作非常敏捷。

凭借这封推荐信，拉马努金 被任命为文书，并于1912年3月1日开始履职。拉马努金 相当幸运，周围有许多受过数学训练的人。事实上，马德拉斯港务信托局的总会计师 S N Aiyar 接受过数学家的训练，并于1913年发表了一篇关于 拉马努金 工作的论文On the distribution of primes。马德拉斯工程学院土木工程教授 C L T Griffith 也对 拉马努金 的能力感兴趣，他曾在伦敦大学学院接受教育，认识那里的数学教授，即 M J M Hill。他于1912年11月12日写信给 Hill，寄去了 拉马努金 的一些工作以及他1911年关于伯努利数的论文副本。

Hill 以相当鼓励的方式回复，但表明他未能理解 拉马努金 关于发散级数的结果。建议 拉马努金 阅读 布罗米奇 的 Theory of infinite series 并没有让 拉马努金 很高兴。拉马努金 写信给 E W 欧内斯特·威廉·霍布森 和 H F 亨利·弗雷德里克·贝克，试图让他们对他的结果感兴趣，但两人都没有回复。1913年1月，拉马努金 在看过 戈弗雷·哈罗德·哈代 1910年著作 Orders of infinity 的副本后，写信给 戈弗雷·哈罗德·哈代。在 拉马努金 给 戈弗雷·哈罗德·哈代 的信中，他介绍了自己和自己的工作[10]：-

我没有上过大学，但接受过普通的学校课程。离开学校后，我一直利用可支配的业余时间研究数学。我没有按大学课程所遵循的传统常规路线走，而是为自己开辟了一条新路。我对发散级数作了一般性的专门研究，我得到的结果被当地数学家称为“令人震惊的”。

戈弗雷·哈罗德·哈代与利特尔伍德一起研究了拉马努金随信附来的那一长串未经证明的定理。2月8日，他回复了拉马努金[3]，信的开头是：

你的来信和你陈述的定理让我极感兴趣。然而你会明白，在我能恰当判断你所做工作的价值之前，我必须看到你某些断言的证明。在我看来，你的结果大致可分为三类：
(1) 有许多结果已经为人所知，或容易从已知定理推出；
(2) 有些结果据我所知是新的且有趣的，但其有趣更多在于其奇特性与表面上的困难，而非其重要性；
(3) 有些结果看来是新的且重要的……

拉马努金对戈弗雷·哈罗德·哈代的回复感到高兴，当他再次写信时，他说[8]：

我在你身上找到了一位同情我工作的朋友。……我已经是一个半饥半饱的人。为了保住我的脑子，我需要食物，这是我首先要考虑的事。你任何一封表示同情的信，都会有助于我在这里从大学或政府获得奖学金。

事实上，马德拉斯大学确实在1913年5月给了拉马努金一项为期两年的奖学金，而在1914年，戈弗雷·哈罗德·哈代把拉马努金带到剑桥大学三一学院，开始了一段非凡的合作。促成这件事并不容易。拉马努金是一位正统的婆罗门，因此是严格的素食者。他的宗教本应阻止他旅行，但这个困难被克服了，部分靠的是E H Neville的努力，他是戈弗雷·哈罗德·哈代在三一学院的同事，在印度讲学时遇到了拉马努金。

拉马努金于1914年3月17日从印度启航。除了有三天拉马努金晕船外，航行很平静。他于1914年4月14日抵达伦敦，由Neville迎接。在伦敦待了四天后，他们去了剑桥，拉马努金在Neville家中住了几周，然后于4月30日搬入三一学院的房间。然而，从一开始，他的饮食就有问题。第一次世界大战的爆发使获得特殊食品变得更加困难，不久拉马努金就出现了健康问题。

从一开始，拉马努金与戈弗雷·哈罗德·哈代的合作就产生了重要成果。然而，戈弗雷·哈罗德·哈代不确定如何应对拉马努金缺乏正规教育的问题。他写道[1]：-

在教授他现代数学方面该怎么办？他知识的局限性和其深度一样令人震惊。

利特尔伍德被要求帮助教授拉马努金严格的数学方法。然而，他说（[31]）：-

……这极其困难，因为每当提到一些被认为拉马努金需要知道的事情时，拉马努金的回应就是大量原创想法，这使得利特尔伍德几乎不可能坚持他最初的意图。

战争很快让利特尔伍德去服兵役，但戈弗雷·哈罗德·哈代留在剑桥与拉马努金一起工作。即使在他于英格兰的第一个冬天，拉马努金也生病了，他在1915年3月写道，由于冬季天气他生病了，并且五个月来无法发表任何东西。他确实发表的是他在英格兰所做的工作，已经决定他在印度期间获得的结果，其中许多他已在信中告知戈弗雷·哈罗德·哈代，将不会发表，直到战争结束。

1916年3月16日，拉马努金以研究型文学学士从剑桥毕业（该学位自1920年起称为Ph.D.）。尽管没有适当的资格，他还是在1914年6月获准入学。拉马努金的学位论文是关于Highly composite numbers的，由他在英格兰发表的七篇论文组成。

拉马努金在1917年重病，他的医生担心他会死。到9月他确实有所好转，但大部分时间都在各种疗养院度过。1918年2月，戈弗雷·哈罗德·哈代写道（见[3]）：-

Batty Shaw发现了一件其他医生都不知道的事：他大约四年前做过一次手术。他最坏的理论是，那次手术实际上是为了切除一个被误诊的恶性肿瘤。鉴于拉马努金并不比六个月前更差，他现在已经放弃了这个理论——其他医生从未支持过它。自从最初认为的胃溃疡被放弃以来，结核一直是暂时被接受的理论。……像所有印度人一样，他是宿命论者，要让他照顾自己极其困难。

1918年2月18日，拉马努金当选为剑桥哲学学会会士，三天后，他将获得的最高荣誉——他的名字出现在伦敦皇家学会会士选举名单上。他由一批令人印象深刻的数学家提名，即戈弗雷·哈罗德·哈代、珀西·亚历山大·麦克马洪、Grace、约瑟夫·拉莫尔、布罗米奇、欧内斯特·威廉·霍布森、亨利·弗雷德里克·贝克、利特尔伍德、Nicholson、威廉·亨利·扬、埃德蒙·泰勒·惠特克、安德鲁·福赛思和阿尔弗雷德·诺思·怀特海。他当选为皇家学会会士于1918年5月2日得到确认，随后在1918年10月10日，他当选为剑桥大学三一学院研究员，该研究员职位为期六年。

授予拉马努金的荣誉似乎帮助他的健康略有改善，他重新努力研究数学。到1918年11月底，拉马努金的健康已大大改善。戈弗雷·哈罗德·哈代在一封信中写道[3]：-

我想我们现在可以希望他已经渡过难关，正在走向真正的康复。他的体温已不再不规则，体重增加了近一英石。……他的非凡数学才能从未有任何减退的迹象。自然，在患病期间他产出较少，但质量一直相同。……

他将带着前所未有的印度人所享有的科学地位和声誉返回印度，我相信印度会把他视为他本是的珍宝。他天生的纯朴和谦逊从未因成功而受到丝毫影响——事实上，所需要的只是让他意识到他确实是一个成功者。

拉马努金于1919年2月27日乘船前往印度，于3月13日抵达。然而，他的健康状况非常差，尽管接受了治疗，他还是在次年在那里去世。

拉马努金在1913年写给戈弗雷·哈罗德·哈代的信中包含了许多引人入胜的结果。拉马努金算出了波恩哈德·黎曼级数、椭圆积分、超几何级数以及ζ函数的函数方程。另一方面，他对什么构成数学证明只有模糊的概念。尽管有许多辉煌的结果，他关于素数的一些定理是完全错误的。

拉马努金独立发现了卡尔·弗里德里希·高斯、恩斯特·爱德华·库默尔等人关于超几何级数的结果。拉马努金自己关于超几何级数的部分和与乘积的工作导致了该主题的重大发展。也许他最著名的工作是关于整数$n$分拆成被加数所得的分拆数$p(n)$。珀西·亚历山大·麦克马洪制作了小数字$n$的$p(n)$值表，拉马努金利用这些数值数据猜想了一些显著的性质，其中一些他用椭圆函数证明了。其他的直到拉马努金去世后才被证明。

在与戈弗雷·哈罗德·哈代合写的一篇论文中，拉马努金给出了$p(n)$的渐近公式。它具有一个显著的性质：它似乎给出了$p(n)$的正确值，这后来由汉斯·拉代马海尔证明。

拉马努金留下了许多未发表的手稿，里面充满了数学家们持续研究的定理。乔治·奈维尔·沃森，1918年至1951年间任伯明翰纯数学梅森讲席教授，以Theorems stated by Ramanujan为总标题发表了14篇论文，总共发表了近30篇受拉马努金工作启发的论文。戈弗雷·哈罗德·哈代将他所拥有的大量拉马努金的手稿传给了乔治·奈维尔·沃森，这些手稿既有1914年之前写的，也有拉马努金在印度去世前最后一年写的。

上图取自印度邮政局为庆祝他诞辰75周年而发行的一枚邮票。

Srinivasa Ramanujan was one of India's greatest mathematical geniuses. He made substantial contributions to the analytical theory of numbers and worked on elliptic functions, continued fractions, and infinite series.

Ramanujan was born in his grandmother's house in Erode, a small village about 400 km southwest of Madras (now Chennai). When Ramanujan was a year old his mother took him to the town of Kumbakonam, about 160 km nearer Madras. His father worked in Kumbakonam as a clerk in a cloth merchant's shop. In December 1889 he contracted smallpox.

When he was nearly five years old, Ramanujan entered the primary school in Kumbakonam although he would attend several different primary schools before entering the Town High School in Kumbakonam in January 1898. At the Town High School, Ramanujan was to do well in all his school subjects and showed himself an able all-round scholar. In 1900 he began to work on his own on mathematics summing geometric and arithmetic series.

Ramanujan was shown how to solve cubic equations in 1902 and he went on to find his own method to solve the quartic. The following year, not knowing that the quintic could not be solved by radicals, he tried (and of course failed) to solve the quintic.

It was in the Town High School that Ramanujan came across a mathematics book by G S Carr called Synopsis of elementary results in pure mathematics. This book, with its very concise style, allowed Ramanujan to teach himself mathematics, but the style of the book was to have a rather unfortunate effect on the way Ramanujan was later to write down mathematics since it provided the only model that he had of written mathematical arguments. The book contained theorems, formulae and short proofs. It also contained an index to papers on pure mathematics which had been published in the European Journals of Learned Societies during the first half of the 19th century. The book, published in 1886, was of course well out of date by the time Ramanujan used it.

By 1904 Ramanujan had begun to undertake deep research. He investigated the series $\sum (\Large\frac{1}{n}\normalsize )$ and calculated Euler's constant to 15 decimal places. He began to study the Bernoulli numbers, although this was entirely his own independent discovery.

Ramanujan, on the strength of his good school work, was given a scholarship to the Government College in Kumbakonam which he entered in 1904. However, the following year his scholarship was not renewed because Ramanujan devoted more and more of his time to mathematics and neglected his other subjects. Without money, he was soon in difficulties and, without telling his parents, he ran away to the town of Vizagapatnam about 650 km north of Madras. He continued his mathematical work, however, and at this time he worked on hypergeometric series and investigated relations between integrals and series. He was to discover later that he had been studying elliptic functions.

In 1906 Ramanujan went to Madras where he entered Pachaiyappa's College. His aim was to pass the First Arts examination which would allow him to be admitted to the University of Madras. He attended lectures at Pachaiyappa's College but became ill after three month's study. He took the First Arts examination after having left the course. He passed in mathematics but failed all his other subjects and therefore failed the examination. This meant that he could not enter the University of Madras. In the following years he worked on mathematics developing his own ideas without any help and without any real idea of the then current research topics other than that provided by Carr's book.

Continuing his mathematical work, Ramanujan studied continued fractions and divergent series in 1908. At this stage he became seriously ill again and underwent an operation in April 1909 after which it took him some considerable time to recover. He married on 14 July 1909 when his mother arranged for him to marry a ten-year-old girl S Janaki Ammal. Ramanujan did not live with his wife, however, until she was twelve years old.

Ramanujan continued to develop his mathematical ideas and began to pose problems and solve problems in the Journal of the Indian Mathematical Society. He developed relations between elliptic modular equations in 1910. After publication of a brilliant research paper on Bernoulli numbers in 1911 in the Journal of the Indian Mathematical Society, he gained recognition for his work. Despite his lack of a university education, he was becoming well known in the Madras area as a mathematical genius.

In 1911 Ramanujan approached the founder of the Indian Mathematical Society for advice on a job. After this he was appointed to his first job, a temporary post in the Accountant General's Office in Madras. It was then suggested that he approach Ramachandra Rao who was a Collector at Nellore. Ramachandra Rao was a founder member of the Indian Mathematical Society who had helped start the mathematics library. He writes in [30]:-

A short uncouth figure, stout, unshaven, not over clean, with one conspicuous feature – shining eyes – walked in with a frayed notebook under his arm. He was miserably poor. ... He opened his book and began to explain some of his discoveries. I saw quite at once that there was something out of the way; but my knowledge did not permit me to judge whether he talked sense or nonsense. ... I asked him what he wanted. He said he wanted a pittance to live on so that he might pursue his researches.

Ramachandra Rao told him to return to Madras and he tried, unsuccessfully, to arrange a scholarship for Ramanujan. In 1912 Ramanujan applied for the post of clerk in the accounts section of the Madras Port Trust. In his letter of application he wrote [3]:-

I have passed the Matriculation Examination and studied up to the First Arts but was prevented from pursuing my studies further owing to several untoward circumstances. I have, however, been devoting all my time to Mathematics and developing the subject.

Despite the fact that he had no university education, Ramanujan was clearly well known to the university mathematicians in Madras for, with his letter of application, Ramanujan included a reference from E W Middlemast who was the Professor of Mathematics at The Presidency College in Madras. Middlemast, a graduate of St John's College, Cambridge, wrote [3]:-

I can strongly recommend the applicant. He is a young man of quite exceptional capacity in mathematics and especially in work relating to numbers. He has a natural aptitude for computation and is very quick at figure work.

On the strength of the recommendation Ramanujan was appointed to the post of clerk and began his duties on 1 March 1912. Ramanujan was quite lucky to have a number of people working round him with a training in mathematics. In fact the Chief Accountant for the Madras Port Trust, S N Aiyar, was trained as a mathematician and published a paper On the distribution of primes in 1913 on Ramanujan's work. The professor of civil engineering at the Madras Engineering College C L T Griffith was also interested in Ramanujan's abilities and, having been educated at University College London, knew the professor of mathematics there, namely M J M Hill. He wrote to Hill on 12 November 1912 sending some of Ramanujan's work and a copy of his 1911 paper on Bernoulli numbers.

Hill replied in a fairly encouraging way but showed that he had failed to understand Ramanujan's results on divergent series. The recommendation to Ramanujan that he read Bromwich's Theory of infinite series did not please Ramanujan much. Ramanujan wrote to E W Hobson and H F Baker trying to interest them in his results but neither replied. In January 1913 Ramanujan wrote to G H Hardy having seen a copy of his 1910 book Orders of infinity. In Ramanujan's letter to Hardy he introduced himself and his work [10]:-

I have had no university education but I have undergone the ordinary school course. After leaving school I have been employing the spare time at my disposal to work at mathematics. I have not trodden through the conventional regular course which is followed in a university course, but I am striking out a new path for myself. I have made a special investigation of divergent series in general and the results I get are termed by the local mathematicians as 'startling'.

Hardy, together with Littlewood, studied the long list of unproved theorems which Ramanujan enclosed with his letter. On 8 February he replied to Ramanujan [3], the letter beginning:-

I was exceedingly interested by your letter and by the theorems which you state. You will however understand that, before I can judge properly of the value of what you have done, it is essential that I should see proofs of some of your assertions. Your results seem to me to fall into roughly three classes:
(1)  there are a number of results that are already known, or easily deducible from known theorems;
(2)  there are results which, so far as I know, are new and interesting, but interesting rather from their curiosity and apparent difficulty than their importance;
(3)  there are results which appear to be new and important...

Ramanujan was delighted with Hardy's reply and when he wrote again he said [8]:-

I have found a friend in you who views my labours sympathetically. ... I am already a half starving man. To preserve my brains I want food and this is my first consideration. Any sympathetic letter from you will be helpful to me here to get a scholarship either from the university or from the government.

Indeed the University of Madras did give Ramanujan a scholarship in May 1913 for two years and, in 1914, Hardy brought Ramanujan to Trinity College, Cambridge, to begin an extraordinary collaboration. Setting this up was not an easy matter. Ramanujan was an orthodox Brahmin and so was a strict vegetarian. His religion should have prevented him from travelling but this difficulty was overcome, partly by the work of E H Neville who was a colleague of Hardy's at Trinity College and who met with Ramanujan while lecturing in India.

Ramanujan sailed from India on 17 March 1914. It was a calm voyage except for three days on which Ramanujan was seasick. He arrived in London on 14 April 1914 and was met by Neville. After four days in London they went to Cambridge and Ramanujan spent a couple of weeks in Neville's home before moving into rooms in Trinity College on 30th April. Right from the beginning, however, he had problems with his diet. The outbreak of World War I made obtaining special items of food harder and it was not long before Ramanujan had health problems.

Right from the start Ramanujan's collaboration with Hardy led to important results. Hardy was, however, unsure how to approach the problem of Ramanujan's lack of formal education. He wrote [1]:-

What was to be done in the way of teaching him modern mathematics? The limitations of his knowledge were as startling as its profundity.

Littlewood was asked to help teach Ramanujan rigorous mathematical methods. However, he said ([31]):-

... that it was extremely difficult because every time some matter, which it was thought that Ramanujan needed to know, was mentioned, Ramanujan's response was an avalanche of original ideas which made it almost impossible for Littlewood to persist in his original intention.

The war soon took Littlewood away on war duty but Hardy remained in Cambridge to work with Ramanujan. Even in his first winter in England, Ramanujan was ill and he wrote in March 1915 that he had been ill due to the winter weather and had not been able to publish anything for five months. What he did publish was the work he did in England, the decision having been made that the results he had obtained while in India, many of which he had communicated to Hardy in his letters, would not be published until the war had ended.

On 16 March 1916 Ramanujan graduated from Cambridge with a Bachelor of Arts by Research (the degree was called a Ph.D. from 1920). He had been allowed to enrol in June 1914 despite not having the proper qualifications. Ramanujan's dissertation was on Highly composite numbers and consisted of seven of his papers published in England.

Ramanujan fell seriously ill in 1917 and his doctors feared that he would die. He did improve a little by September but spent most of his time in various nursing homes. In February 1918 Hardy wrote (see [3]):-

Batty Shaw found out, what other doctors did not know, that he had undergone an operation about four years ago. His worst theory was that this had really been for the removal of a malignant growth, wrongly diagnosed. In view of the fact that Ramanujan is no worse than six months ago, he has now abandoned this theory - the other doctors never gave it any support. Tubercle has been the provisionally accepted theory, apart from this, since the original idea of gastric ulcer was given up. ... Like all Indians he is fatalistic, and it is terribly hard to get him to take care of himself.

On 18 February 1918 Ramanujan was elected a fellow of the Cambridge Philosophical Society and then three days later, the greatest honour that he would receive, his name appeared on the list for election as a fellow of the Royal Society of London. He had been proposed by an impressive list of mathematicians, namely Hardy, MacMahon, Grace, Larmor, Bromwich, Hobson, Baker, Littlewood, Nicholson, Young, Whittaker, Forsyth and Whitehead. His election as a fellow of the Royal Society was confirmed on 2 May 1918, then on 10 October 1918 he was elected a Fellow of Trinity College Cambridge, the fellowship to run for six years.

The honours which were bestowed on Ramanujan seemed to help his health improve a little and he renewed his effort at producing mathematics. By the end of November 1918 Ramanujan's health had greatly improved. Hardy wrote in a letter [3]:-

I think we may now hope that he has turned the corner, and is on the road to a real recovery. His temperature has ceased to be irregular, and he has gained nearly a stone in weight. ... There has never been any sign of any diminution in his extraordinary mathematical talents. He has produced less, naturally, during his illness but the quality has been the same. ....

He will return to India with a scientific standing and reputation such as no Indian has enjoyed before, and I am confident that India will regard him as the treasure he is. His natural simplicity and modesty have never been affected in the least by success - indeed all that is wanted is to get him to realise that he really is a success.

Ramanujan sailed to India on 27 February 1919 arriving on 13 March. However, his health was very poor and, despite medical treatment, he died there the following year.

The letters Ramanujan wrote to Hardy in 1913 had contained many fascinating results. Ramanujan worked out the Riemann series, the elliptic integrals, hypergeometric series and functional equations of the zeta function. On the other hand he had only a vague idea of what constitutes a mathematical proof. Despite many brilliant results, some of his theorems on prime numbers were completely wrong.

Ramanujan independently discovered results of Gauss, Kummer and others on hypergeometric series. Ramanujan's own work on partial sums and products of hypergeometric series has led to major development in the topic. Perhaps his most famous work was on the number $p(n)$ of partitions of an integer $n$ into summands. MacMahon had produced tables of the value of $p(n)$ for small numbers $n$, and Ramanujan used this numerical data to conjecture some remarkable properties some of which he proved using elliptic functions. Others were only proved after Ramanujan's death.

In a joint paper with Hardy, Ramanujan gave an asymptotic formula for $p(n)$. It had the remarkable property that it appeared to give the correct value of $p(n)$, and this was later proved by Rademacher.

Ramanujan left a number of unpublished notebooks filled with theorems that mathematicians have continued to study. G N Watson, Mason Professor of Pure Mathematics at Birmingham from 1918 to 1951 published 14 papers under the general title Theorems stated by Ramanujan and in all he published nearly 30 papers which were inspired by Ramanujan's work. Hardy passed on to Watson the large number of manuscripts of Ramanujan that he had, both written before 1914 and some written in Ramanujan's last year in India before his death.

The picture above is taken from a stamp issued by the Indian Post Office to celebrate the 75th anniversary of his birth.