Ramanujan's sum

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

${\displaystyle c_{q}(n)=\sum _{1\leq a\leq q \atop (a,q)=1}e^{2\pi i{\tfrac {a}{q}}n},}$

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

Srinivasa Ramanujan mentioned 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.