Knowra Srinivasa Ramanujan Srinivasa Ramanujan Srinivasa Ramanujan was an Indian mathematician whose work in number theory, infinite series, and continued fractions produced results that continue to shape modern mathematics.
Number theory : The branch of mathematics that studies integers and their properties. Its questions about primes, partitions, and divisibility run through Ramanujan’s research.
G. H. Hardy : G. H. Hardy was a British mathematician known for work in analysis and number theory. Hardy recognized Ramanujan’s talent and became his collaborator at Cambridge.
Hardy–Ramanujan asymptotic formula : An asymptotic formula estimates the partition function p(n) as n grows, with leading term proportional to exp(π√(2n/3))/(4n√3). Ramanujan and Hardy derived this striking estimate for the number of integer partitions.
Chudnovsky algorithm : A rapidly converging series algorithm for calculating digits of pi, derived from hypergeometric series related to Ramanujan’s formulas. It extends the strategy of Ramanujan-type series to high-precision computation of pi.
Lost notebook : The Lost Notebook is a collection of Ramanujan’s mathematical papers found in 1976 among the papers of G. N. Watson. Its many mock theta function identities renewed research into his late work.
Partition function : The function p(n) counts the ways a positive integer can be written as a sum of positive integers, disregarding order. Ramanujan discovered striking congruences and asymptotic formulas for this function.
J. E. Littlewood : J. E. Littlewood was a British mathematician whose work spanned analysis and number theory. Littlewood and Hardy helped establish the importance of Ramanujan’s results.
Ramanujan congruences : Congruences showing that p(5n+4), p(7n+5), and p(11n+6) are divisible by 5, 7, and 11, respectively. These divisibility patterns exemplify the unexpected arithmetic structure he found in partitions.
Black hole entropy : A measure of the number of microscopic states associated with a black hole, proportional to its horizon area. Mock modular forms linked to Ramanujan help count certain black-hole states.
Mock theta function : A q-series resembling a theta function in its transformation behavior but lacking full modularity on its own. Ramanujan introduced these functions near the end of his life, leaving their structure unresolved.
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