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Ramanujan's sum. In number theory, Ramanujan's sum, usually denoted cq ( 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 . Srinivasa Ramanujan mentioned the sums in a 1918 paper. [ 1]
Srinivasa Ramanujan[ a] (22 December 1887 – 26 April 1920) was an Indian mathematician. 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 considered unsolvable.
In mathematics, the Ramanujan conjecture, due to Srinivasa Ramanujan ( 1916 , p. 176), states that Ramanujan's tau function given by the Fourier coefficients τ(n) of the cusp form Δ (z) of weight 12. where , satisfies. when p is a prime number.
In number theory, the partition function p(n) represents the number of possible partitions of a non-negative integer n. For instance, p(4) = 5 because the integer 4 has the five partitions 1 + 1 + 1 + 1, 1 + 1 + 2, 1 + 3, 2 + 2, and 4 . No closed-form expression for the partition function is known, but it has both asymptotic expansions that ...
1729 (number) 1729 is the natural number following 1728 and preceding 1730. It is the first nontrivial taxicab number, expressed as the sum of two cubic numbers in two different ways. It is also known as the Ramanujan number or Hardy–Ramanujan number, named after G. H. Hardy and Srinivasa Ramanujan .
In number theory and combinatorics, a partition of a non-negative integer n, also called an integer partition, is a way of writing n as a sum of positive integers. Two sums that differ only in the order of their summands are considered the same partition. (If order matters, the sum becomes a composition .)
Ramanujan–Sato series. 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.
Divisor functions were studied by Ramanujan, who gave a number of important congruences and identities; these are treated separately in the article Ramanujan's sum. A related function is the divisor summatory function, which, as the name implies, is a sum over the divisor function.