Ramanujan's Sum

In number theory, a branch of mathematics, Ramanujan's sum, usually denoted cq(n), is a function of two positive integer variables q and n defined by the formula

c_q(n)= \sum_{a=1\atop (a,q)=1}^q e^{2 \pi i \tfrac{a}{q} n} ,

where (a, q) = 1 means that a only takes on values coprime to q.

Srinivasa Ramanujan introduced the sums in a 1918 paper. 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.

Read more about Ramanujan's Sum:  Notation, Table, Ramanujan Expansions, See Also

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