Srinivasa Ramanujan: The Nobel Prize winner

Srinivasa Ramanujan(22 December 1887 – 26 April 1920) was an Indian mathematician, who, with almost no formal training in pure mathematics, made substantial contributions to mathematical analysis, number theory, infinite series and continued fractions.Born and raised in Erode, Tamil Nadu, India, Ramanujan first encountered formal mathematics at age ten. He demonstrated a natural ability, and was given books on advanced trigonometry by S. L. Loney. He had mastered them by age thirteen, and even discovered theorems of his own. He demonstrated unusual mathematical skills at school, winning accolades and awards. By seventeen, Ramanujan conducted his own mathematical research on Bernoulli numbers and the Euler–Mascheroni constant. He received a scholarship to study at Government College in Kumbakonam, but lost it when he failed his non-mathematical coursework. He joined another college to pursue independent mathematical research, working as a clerk in the Accountant-General's office at the Madras Port Trust Office to support himself. In 1912-1913, he sent samples of his theorems to three academicians at the University of Cambridge. Only G. H. Hardy recognized the brilliance of his work, subsequently inviting Ramanujan to study under him at Cambridge.

Ramanujan independently compiled nearly 3900 results (mostly identities and equations) during his short lifetime. Although a small number of these results were actually false and some were already known, most of his claims have now been proven correct. He stated results that were both original and highly unconventional, such as the Ramanujan prime and the Ramanujan theta function, and these have inspired a vast amount of further research. However, some of his major discoveries have been rather slow to enter the mathematical mainstream. Recently, Ramanujan's formulae have found applications in crystallography and string theory. The Ramanujan Journal, an international publication, was launched to publish work in all areas of mathematics influenced by his work.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. It is said that Ramanujan's discoveries are unusually rich and that there is often more in it than what initially meets the eye. As a by-product, new directions of research were opened up. Examples of the most interesting of these formulae include the intriguing infinite series for π, one of which is given below

This result is based on the negative fundamental discriminant d = −4×58 with class number h(d) = 2 (note that 5×7×13×58 = 26390) and is related to the fact that.Compare to Heegner numbers, which have class number 1 and yield similar formulae. Ramanujan's series for π converges extraordinarily rapidly (exponentially) and forms the basis of some of the fastest algorithms currently used to calculate π. Truncating the sum to the first term also gives the approximation for π, which is correct to six decimal places.

One of his remarkable capabilities was the rapid solution for problems. He was sharing a room with P. C. Mahalanobis who had a problem, "Imagine that you are on a street with houses marked 1 through n. There is a house in between (x) such that the sum of the house numbers to left of it equals the sum of the house numbers to its right. If n is between 50 and 500, what are n and x." This is a bivariate problem with multiple solutions. Ramanujan thought about it and gave the answer with a twist: He gave a continued fraction. The unusual part was that it was the solution to the whole class of problems. Mahalanobis was astounded and asked how he did it. "It is simple. The minute I heard the problem, I knew that the answer was a continued fraction. Which continued fraction, I asked myself. Then the answer came to my mind", Ramanujan replied.His intuition also led him to derive some previously unknown identities, such as

for all θ, where Γ(z) is the gamma function. Equating coefficients of θ0, θ4, and θ8 gives some deep identities for the hyperbolic secant.In 1918, G. H. Hardy and Ramanujan studied the partition function P(n) extensively and gave a very accurate non-convergent asymptotic series that permits exact computation of the number of partitions of an integer. Hans Rademacher, in 1937, was able to refine their formula to find an exact convergent series solution to this problem. Ramanujan and Hardy's work in this area gave rise to a powerful new method for finding asymptotic formulae, called the circle method.He discovered mock theta functions in the last year of his life. For many years these functions were a mystery, but they are now known to be the holomorphic parts of harmonic weak Maass forms.