Ramanujan's Sum

Ramanujans sum, written c sub q of n, is a function of two positive integers q and n defined in number theory as the sum, over every integer a from 1 to q that is coprime to q, of the exponential term e raised to 2 pi i times a over q times n. Srinivasa Ramanujan introduced these sums in a 1918 paper, remarking that they seemed to be of great interest and appeared not to have been studied before. The sums always take integer values, are multiplicative as functions of q for a fixed n, and satisfy orthogonality relationships that make them useful throughout analytic number theory, including in the study of divisor functions, the distribution of primes, and the representation of integers as sums of squares or triangular numbers. Ramanujans sums also play a key role in proving Vinogradovs theorem, which shows that every sufficiently large odd number can be written as the sum of three primes.