Ramanujan's Sum

In number theory, a branch of mathematics, 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.

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.