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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]
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.
The 1911 volume of the Journal contains one of the earliest contributions of the Indian mathematician Srinivasa Ramanujan. It was in the form of a set of questions. A fifteen page paper entitled Some properties of Bernoulli Numbers [1] contributed by Ramanujan also appeared in the same 1911 volume of the Journal.
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Ramanujan summation is a method to isolate the constant term in the Euler–Maclaurin formula for the partial sums of a series. For a function f , the classical Ramanujan sum of the series ∑ k = 1 ∞ f ( k ) {\displaystyle \textstyle \sum _{k=1}^{\infty }f(k)} is defined as
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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 ...