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.

For integers a and b, ${\displaystyle a\mid b}$ is read "a divides b" and means that there is an integer c such that b = ac. Similarly, ${\displaystyle a\nmid b}$ is read "a does not divide b". The summation symbol

means that d goes through all the positive divisors of m, e.g.

${\displaystyle (a,\,b)\;}$ is the greatest common divisor,

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