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In mathematics, a character sum is a sum () of values of a Dirichlet character χ modulo N, taken over a given range of values of n.Such sums are basic in a number of questions, for example in the distribution of quadratic residues, and in particular in the classical question of finding an upper bound for the least quadratic non-residue modulo N.
Srinivasa Ramanujan mentioned the sums in a 1918 paper. [1] In addition to the expansions discussed in this article, Ramanujan's sums are used in the proof of Vinogradov's theorem that every sufficiently large odd number is the sum of three primes. [2]
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.
In the meantime, Ramanujan is diagnosed with tuberculosis and his frequent letters home to his wife remain unanswered after many months. Hardy continues to see much more promise in Ramanujan. However, he remains unaware of the personal difficulties his student is having with his housing and with his lack of contact with his family back home in ...
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Ramanujan summation is a technique invented by the mathematician Srinivasa Ramanujan for assigning a value to divergent infinite series.Although the Ramanujan summation of a divergent series is not a sum in the traditional sense, it has properties that make it mathematically useful in the study of divergent infinite series, for which conventional summation is undefined.
New York Rangers forward Matt Rempe was suspended for eight games by NHL’s Department of Player Safety on Sunday for boarding and elbowing Dallas Stars defenseman Miro Heiskanen on Friday night.
Saraswathi of Rediff wrote, "Ramanujan is a brilliant film, a must watch" and gave the film 4/5. [20] Gautaman Bhaskaran of The Hindustan Times gave the film 3/5 stars and wrote, "The movie is a poignant look at the way a prodigy struggled and suffered in a penurious family, a mastermind whose mathematical wizardry invited ridicule and ...