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In mathematics, Ramanujan's master theorem, named after Srinivasa Ramanujan, [1] is a technique that provides an analytic expression for the Mellin transform of an analytic function. Page from Ramanujan's notebook stating his Master theorem.
Srinivasa Ramanujan Aiyangar [a] (22 December 1887 – 26 April 1920) was an Indian mathematician.Often regarded as one of the greatest mathematicians of all time, though he had almost no formal training in pure mathematics, he made substantial contributions to mathematical analysis, number theory, infinite series, and continued fractions, including solutions to mathematical problems then ...
If is a Ramanujan graph, then is a bipartite Ramanujan graph, so the existence of Ramanujan graphs is stronger. As observed by Toshikazu Sunada , a regular graph is Ramanujan if and only if its Ihara zeta function satisfies an analog of the Riemann hypothesis .
In number theory, Ramanujan's sum, usually denoted c q (n), is a function of two positive integer variables q and n defined by the formula = (,) =, where (a, q) = 1 means that a only takes on values coprime to q.
The multiplicative inverse of its generating function is the Euler function; by Euler's pentagonal number theorem this function is an alternating sum of pentagonal number powers of its argument. Srinivasa Ramanujan first discovered that the partition function has nontrivial patterns in modular arithmetic, now known as Ramanujan's congruences.
Ramanujam vanishing theorem (algebraic geometry) Ramanujan–Skolem's theorem (Diophantine equations) Ramsey's theorem (graph theory, combinatorics) Rank–nullity theorem (linear algebra) Rao–Blackwell theorem ; Rashevsky–Chow theorem (control theory) Rational root theorem (algebra, polynomials) Rationality theorem
The initial idea is usually attributed to the work of Hardy with Srinivasa Ramanujan a few years earlier, in 1916 and 1917, on the asymptotics of the partition function.It was taken up by many other researchers, including Harold Davenport and I. M. Vinogradov, who modified the formulation slightly (moving from complex analysis to exponential sums), without changing the broad lines.
In mathematics, a Ramanujan–Sato series [1] [2] generalizes Ramanujan’s pi formulas such as, = = ()!! + to the form = = + by using other well-defined sequences of integers obeying a certain recurrence relation, sequences which may be expressed in terms of binomial coefficients (), and ,, employing modular forms of higher levels.