Srinivasa Ramanujan

Srinivasa Aiyangar Ramanujan (Tamil: ஸ்ரீனிவாஸ ஐயங்கார் ராமானுஜன்) (December 22, 1887 – April 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".

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,

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

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which is related to the fact that,

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

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Hardy wrote of Ramanujan:

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:"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..."

Related Topics:
Modular equation - Continued fraction - Zeta function - Cauchy's theorem - Complex variable

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Theorems and discoveries

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

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• Properties of highly composite numbers
• The partition function and its asymptotics
• Mock theta functions

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

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• Gamma functions
• Modular forms
• Divergent series
• Hypergeometric series
• Prime number theory

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

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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.

Related Topics:
Ramanujan conjecture - Cusp form - Modular forms - Weil conjectures

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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.

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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)

Related Topics:
Slate - G. S. Carr

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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)

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