Life of Srinivasa Ramanujan and his contributions in the field of Mathematics

Srinivasa Ramanujan was a self-educated prodigy from a small place named Kumbakonam in Tamil Nadu. He was born into a poor Brahmin Family in 1887. He started his formal education at the age of 10 years. He was very good at advanced trigonometry and mastered it by the age of twelve. His zeal for mathematics made him carry out his research activities independently. By the age of 17, he started to work on various contemporary research areas such as Bernoulli Numbers and Euler's constant.

Because of his excellence in Mathematics, he received a scholarship to one of the eminent institutions namely, Government College, Kumbakonam but he lost it as he was unable to perform up to the mark in other non-mathematical subjects. In order to sustain his livelihood, he worked as a clerk in Accountant-General's Office at the Madras Port Trust office. Around 1912-1913, he managed to somehow send some of his research works to one of the lecturers in Mathematics at Trinity College, Cambridge namely G.H Hardy. Upon viewing the level of brilliance in his works, Hardy decided to call him to trinity in order to work along with him. However, because of his devout Hindu upbringing, he took some time to accept the invitation as crossing the seas was forbidden as per religious beliefs. However, he finally accepted the invitation and this marked the beginning of a new chapter in the life of Ramanujan. 

"Great Knowledge often comes from the Humblest of origins", said John Littlewood to Mr Srinivasa Ramanujan upon his entry into Trinity College, Cambridge upon an invitation received from G.H Hardy(As highlighted in the movie: The man who knew infinity).

 Upon arrival, Ramanujan showed his other works to G.H Hardy and John Littlewood and it was so impressive that John Littlewood said, "It will take a lifetime"(in order to comprehend this). The life of Ramanujan at trinity was not at all easy. Many of the lecturers in trinity, were not happy with the decision of Hardy's invitation to Ramanujan. However backing Ramanujan, Hardy always tried to convince the fellows around and said Ramanujan was a natural genius. Tussles between the rival lecturers, who were quite unwilling to accept Ramanujan at first and constant support from Hardy to Ramanujan, depict the overall scenario of what actually happened at that time in Trinity.  

Ramanujan was born into a brahmin family and his faith and belief in God were quite evident in his talks with Hardy. The internal academic tussles between G.H Hardy and Ramanujan are also to be a point of mention. Ramanujan and Hardy frequently disagreed because of the substantial contrasts in their working methods. Deeply devout, Ramanujan saw his intuition and insights as "divine inspiration," which he used to produce significant discoveries. Ramanujan was difficult for Hardy, an ardent atheist, to persuade of the value of proof and rigour in his work. The two mathematicians nevertheless worked effectively together for five years. Many of the formulae they discovered bear their names, most notably the Hardy-Ramanujan asymptotic formula and the Hardy-Ramanujan theorem. 

Ramanujan was awarded a PhD (then a Bachelor of Science degree by Research) in 1916. The first part of his final thesis was published in the Journal of the London Mathematical Society. In May 1918 he was elected as a Fellow of the Royal Society(FRS), one of the most prestigious academic positions. But the journey of getting awarded an FRS was not at all easy. Hardy's role is to be mentioned here, who was the key to, convincing the judges of the Royal Society by showing the profoundness and originality in Ramanujan's Research work. This incident also displays the bond that both Hardy and Ramanujan shared even though, their beliefs and faiths were contradictory. Hardy was a lifelong bachelor and in his final years of life, he was cared for by his sister. In a lecture on Ramanujan, Hardy once said "My association with him is the one romantic incident in my life". 

Some of the great works that emerged out of the collective efforts of Hardy and Ramanujan:

1. Hardy-Ramanujan theorem: The normal order of the number ω(n) of distinct prime factors of a number n is log(log(n)). (In Number-theory, a normal order of an arithmetic function is some simpler or better-understood function which usually takes the same or closely approximated values)

2. It was shown by Ramanujan that: 

   •    p(5k+4) is divisible by 5

   •    p(7k+5) is divisible by 7

   •    p(11k+6) is divisible by 11

   
   

               where p(n) refers to the number of partitions for any number n.

3. Hardy-Ramanujan asymptotic formula:  

           Here p(n) refers to the number of partitions for any number n.

           As $n\to \infty$ , the error goes to zero. 

 

One of the projects both Hardy and Ramanujan worked on was to devise a formula to approximate the number of partitions for any number n, to which any algebraic solution seemed to be unsurmountable. Hardy and Ramanujan's asymptotic partition formula is beautiful and elegant since it only uses the numbers 1, 2, 3, and 4, as well as the constants and, and e and therefore has a straightforward structure. It is amazing how a simple equation may approximate values of p(n) with a decreasing error rate as the value of n increases.

G.H hardy, being highly impressed by the insights of Ramanujan once asked him “How did you get all this?”, in a reply to which Ramanujan said, “It’s his devi Namagiri who comes and puts formulae in his mouth when he sleeps or prays”. G.H Hardy was an atheist, he said “I don’t believe in anything that I can’t prove”. Responding to which Ramanujan said,” Then you won’t believe in me”.

”An equation for me has no meaning unless it expresses a thought of God”. 

After hearing this from Ramanujan, Hardy responded “I don’t believe in the immemorial wisdom of the East, but I do believe in you”.(Referenced from the movie: The man who knew infinity).

John Littlewood once said, “Every positive integer is one of the Ramanujan’s personal friends”. This seems to be quite evident in his work in the field of Number theory. 

Unfortunately, we lost such a genius at the very young age of 32 years. Ramanujan independently gathered almost 3900 findings during his brief life (mostly identities and equations). Although some of his findings were erroneous and some were already known, most of his assertions have now been demonstrated to be true. His claims about the Ramanujan prime and the Ramanujan theta function, for example, were both novel and very uncommon, and they have sparked a great deal of additional study. An international journal called The Ramanujan Journal was established to publish work in all branches of mathematics that had been influenced by his contributions. Ramanujan's astounding mathematical works are still a source of the claim that "Intuition rules over logic".