# Srinivasa Ramanujan

Srinivasa Aiyangar Ramanujan (Tamil: ஸ்ரீனிவாஸ ஐயங்கார் ராமானுஜன்) (December 22, 1887April 26, 1920) was a groundbreaking Indian mathematician. A child prodigy, he was largely self-taught in mathematics. Ramanujan is considered one of the world's greatest-ever mathematicians, proving over 3,000 theorems. Ramanujan mainly worked in analytical number theory and is famous for many summation formulas involving constants such as π, prime numbers and the partition function. Often, his formulae were stated without proof and were only later proven to be true. His results inspired a large amount of later research and mathematical papers. In 1997 the Ramanujan Journal was launched to publish work "in areas of mathematics influenced by Ramanujan".

## Life

### Childhood and early life

Ramanujan was a Tamilian Indian born in 1887 in Erode, Tamil Nadu, India. In 1898 at age 10, he entered the Town High School in Kumbakonam, where he appears to have first encountered formal mathematics. At 11 he had mastered the mathematical knowledge of the lodgers at his home, both students at the Government College, and was loaned books on advanced trigonometry, which he mastered by 13. His biographer reports that by 14 his genius was beginning to show. Not only did he achieve merit certificates and academic awards throughout his school years, he was assisting the school in the logistics of assigning its 1200 students (each with their own needs) to its 35-odd teachers, completing exams in half the allotted time, was already showing his familiarity with infinite series; his peers at the time later commented "We, including teachers, rarely understood him," and "stood in respectful awe" of him. However, Ramanujan could not concentrate on other subjects and failed his high school exams. At this time in his life, he was also quite poor and was often pushed to the point of starvation.

### Adulthood in India

As he was married, he had to find a job. With the packet of his mathematical calculations, he moved around in the city of Chennai on the look out of a clerical job. He finally got a job and was advised by an Englishman to contact researchers in Cambridge. As clerk in the Chennai Accountant General's Office, Ramanujan desired the luxury to completely focus on mathematics without having to hold a job. He doggedly solicited support from influential Indian individuals and published several papers in Indian mathematical journals, but was unsuccessful in his attempts to foster sponsorship. At this point of time Sir Ashutosh Mukherjee tried to support his cause.

In 1913 Ramanujan enclosed a long list of complex theorems in a letter to three Cambridge academics: H. F. Baker, E. W. Hobson, and G. H. Hardy. Only Hardy, a Fellow of Trinity College, noticed the genius in Ramanujan’s theorems.

Upon reading the initial unsolicited missive by an unknown and untrained Indian mathematician, Hardy and his colleague J.E. Littlewood commented that, “not one [theorem] could have been set in the most advanced mathematical examination in the world.” Although Hardy was one of the pre-eminent mathematicians of the day and an expert in several of the fields Ramanujan wrote about, he added that many of them "defeated me completely; I had never seen anything in the least like them before."

As an example of his results, Ramanujan gave the beautiful continued fraction,

$\sqrt{\phi+2}- \phi = \cfrac{e^{-2 \pi/5}}{1 + \cfrac{e^{-2 \pi}}{1 + \cfrac{e^{-4 \pi}}{1+ \cfrac{e^{-6 \pi}}{1+\,\cdots}}}} = 0.2840...$

among others, where $\phi=(1+ \sqrt{5})/2$ is the golden ratio.

### Life in England

After some initial skepticism, Hardy replied with comments, requesting proofs for some of the discoveries, and began to make plans to bring Ramanujan to England. As an orthodox Brahmin, Ramanujan consulted the astrological data for his journey, because of religious concerns that he would lose his caste by traveling to foreign shores. Ramanujan's mother had a dream in which the family Goddess told her not to stand in the way of her son's travel, and so he made plans accordingly, although he took pains to keep a proper Brahmin lifestyle as far as he could, when he did.

A fruitful collaboration ensued, which Hardy described as "the one romantic incident in my life". Hardy said of Ramanujan's formulas, some of which he could not initially understand, that "a single look at them is enough to show that they could only be written down by a mathematician of the highest class. They must be true, for if they were not true, no one would have had the imagination to invent them." Hardy stated in an interview by Paul Erdős that his own greatest contribution to mathematics was the discovery of Ramanujan, and compared Ramanujan at least to the mathematical giants Euler and Jacobi in terms of genius. Ramanujan was later appointed a Fellow of Trinity, and the highest level of honor in science, a Fellow of the Royal Society (FRS).

Plagued by health problems all his life, in a country far from home, and obsessively involved with his studies, Ramanujan's health worsened in England, perhaps exacerbated by stress, and by the scarcity of vegetarian food during the First World War. He was diagnosed with tuberculosis (Henderson, 1996) and a severe vitamin deficiency, though a 1994 analysis of Ramanujan's medical records and symptoms by Dr. D.A.B Young concluded that it was much more likely he had hepatic amoebiasis, a parasitic infection of the liver. This is also supported by the fact that Ramanujan spent time in Madras, a coastal city where the disease was widespread. It was a difficult disease to diagnose, but once diagnosed was readily curable (Berndt, 1998). He returned to India in 1919 and died soon after in Kumbakonam, his final gift to the world being the discovery of 'mock Theta functions'. His wife S. Janaki Ammal lived outside Chennai (formerly Madras) until her death in 1994. Janaki had been nine when they were married, a fairly common practice in India at the time. (Henderson, 1996)

### Spiritual life

Ramanujan lived as a Tamil Brahmin all his life. Views of his actual beliefs vary: his first Indian biographers described him as rigorously orthodox, whereas G. H. Hardy (a militant atheist) believed him to be essentially agnostic as far as metaphysical matters were concerned.

Hardy reported a statement of Ramanujan's to the effect that all religions are equally correct. Kanigel's biography states that Ramanujan would probably not have shown Hardy his religious side anyway; on the other hand Kanigel paints a generally negative picture of Hardy.

Ramanujan credited his understanding to his family Goddess, Namagiri, and looked to her for inspiration in his work. He often said, "An equation for me has no meaning, unless it represents a thought of God."

## Mathematical achievements

In mathematics, there is a distinction between having an insight and having a proof. Ramanujan's talent suggested a plethora of formulae that could then be investigated in depth later. As a byproduct, new directions of research were opened up. Examples of these formulae were intriguing infinite series for π, one of which is given by,

$\frac{1}{\pi} = \frac{2\sqrt{2}}{9801} \sum^\infty_{k=0} \frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}}$

which is related to the fact that,

$e^{\pi \sqrt{58}} = 396^4 - 104.00000017...$

Hardy wrote of Ramanujan:

"The limitations of his knowledge were as startling as its profundity. Here was a man who could work out modular equations and theorems... to orders unheard of, whose mastery of continued fractions was... beyond that of any mathematician in the world, who had found for himself the functional equation of the Zeta function and the dominant terms of many of the most famous problems in the analytic theory of numbers; and yet he had never heard of a doubly periodic function or of Cauchy's theorem, and had indeed but the vaguest idea of what a function of a complex variable was..."

### Theorems and discoveries

These include both Ramanujan's own discoveries, and those developed or proven in collaboration with Hardy.

He also made major breakthroughs and discoveries in the areas of:

It is said his discoveries were unusually rich; that is, in many of them there was far more than initially met the eye.

### The Ramanujan conjecture and its role

Although there are numerous statements that could bear the name Ramanujan conjecture, there is one in particular that was very influential on later work. That Ramanujan conjecture is an assertion on the size of the coefficients of the tau-function, a typical cusp form in the theory of modular forms. It was finally proved as a consequence of the proof of the Weil conjectures some decades later; the reduction step is complicated.

### Ramanujan's notebooks

While he was still in India, Ramanujan recorded many results in three notebooks of loose leaf paper. Results were written up, without their derivations. This is probably the origin of the misconception that Ramanujan was unable to prove his results and simply thought the final result up directly. Berndt, in his review of the notebooks and Ramanujan's work felt that Ramanujan most certainly was able to make the proofs of most of his results, but chose not to.

This style of working may have been for several reasons. Since paper was very expensive, Ramanujan would do most of his work and perhaps his proofs on slate, and then transfer just the results to paper. Using a slate was common for mathematics students in India at the time. He was also quite likely to have been influenced by the style of one of the books he had learned much of his advanced mathematics from G. S. Carr's Synopsis of Pure and Applied Mathematics, used by Carr in his tutoring. It summarised several thousand results, stating them without proofs. Finally, it is possible that Ramanujan considered his workings to be for his personal interest alone; and therefore only recorded the results. (Berndt, 1998)

The first notebook was 351 pages with 16 somewhat organized chapters and some unorganized material. The second notebook had 256 pages in 21 chapters and 100 unorganized pages, with the third notebook containing 33 unorganized pages. The results in his notebooks inspired numerous papers by later mathematicians trying to prove what he had found. Hardy himself created papers exploring material from Ramanujan's work as did G. N. Watson, B. M. Wilson, and Bruce Berndt. (Berndt, 1998)

## Quotes

• "Almost a century after his death, it was said of him, "Ramanujan was a mathematician so great that his name transcends jealousies, the one superlatively great mathematician whom India has produced in the last thousand years. His leaps of intuition confound mathematicians even today, seven decades after his death. His papers are still plumbed for their secrets. His theorems are being applied in areas scarcely imaginable during his lifetime." (quoted from Kanigel's biography, "The Man who knew Infinity", p.3)
• "I remember once going to see [Ramanujan] when he was lying ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. 'No,' he replied, 'it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.'" -G.H. Hardy

## Recognition

Ramanujan's home state of Tamil Nadu celebrates 22nd December (Ramanujan's birthday) as 'State IT Day', memorializing both the man, and his achievements, as a native of Tamil Nadu.

A Prize for young mathematicians from developing countries has been created in the name of Srinivasa Ramanujan by the International Centre for Theoretical Physics (ICTP), in cooperation with IMU, who nominate members of the Prize Committee.

## References

• An overview of Ramanujan's notebooks by Bruce C. Berndt, in Charlemagne and His Heritage: 1200 Years of Civilization and Science in Europe, Volume 2: Mathematical Arts, P. L. Butzer, H. Th. Jongen, and W. Oberschelp, editors, Brepols, Turnhout, 1998, pp. 119-146, (22 pg. pdf file)
• Modern Mathematicians, Harry Henderson, Facts on File Inc., 1996

## Cultural references

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