Srinivasa Ramanujan

 Srinivasa Aiyangar Ramanujan, better known as Srinivasa Iyengar Ramanujan (22 December 1887 – 26 April 1920) was an Indian mathematician and self taught genius 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 the age of 10. He demonstrated a natural ability, and was given books on advanced trigonometry written by S L Loney. He had mastered them by the age of 12, and even discovered theorems of his own. He demonstrated unusual mathematical skills at school, winning accolades and awards. By 17, 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 academics at the University of Cambridge. Only G. H. Hardy recognized the brilliance of his work, subsequently inviting Ramanujan to visit and work with him at Cambridge.

 He became a Fellow of the Royal Society and a Fellow of Trinity College, Cambridge, dying of illness, malnutrition and possibly liver infection in 1920 at the age of 32.

 During his short lifetime, Ramanujan independently compiled nearly 3900 results (mostly identities and equations). 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.

 Personality and spiritual life

 Ramanujan has been described as a person with a somewhat shy and quiet disposition, a dignified man with pleasant manners. He lived a rather spartan life while at Cambridge. Ramanujan's first Indian biographers describe him as rigorously orthodox. Ramanujan credited his acumen 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."

 G. H. Hardy cites Ramanujan as remarking that all religions seemed equally true to him. The atheist Hardy further argued that Ramanujan's religiousness had been overstated—in the point of belief, not practice—by his Indian biographers, and romanticised by Westerners. At the same time, he remarked on Ramanujan's strict observance of vegetarianism.

 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.

 The Ramanujan conjecture

 Although there are numerous statements that could bear the name Ramanujan conjecture, there is one statement that was very influential on later work. In particular, the connection of this conjecture with conjectures of André Weil in algebraic geometry opened up new areas of research. That Ramanujan conjecture is an assertion on the size of the tau function, which has as generating function the discriminant modular form, a typical cusp form in the theory of modular forms. It was finally proven in 1973, as a consequence of Pierre Deligne's proof of the Weil conjectures. The reduction step involved is complicated. Deligne won a Fields Medal in 1978 for his work on Weil conjectures.

 Ramanujan's notebooks

 While still in India, Ramanujan recorded the bulk of his results in four notebooks of loose leaf paper. These results were mostly written up without any derivations. This is probably the origin of the misperception that Ramanujan was unable to prove his results and simply thought up the final result directly. Mathematician Bruce C. Berndt, in his review of these notebooks and Ramanujan's work, says 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 G. S. Carr's book, which stated results without proofs. Finally, it is possible that Ramanujan considered his workings to be for his personal interest alone; and therefore only recorded the results.

 The first notebook has 351 pages with 16 somewhat organized chapters and some unorganized material. The second notebook has 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. A fourth notebook, the so-called "lost notebook", was rediscovered in 1976 by George Andrews.