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:These sums are obviously of great interest, and a few of their properties have been discussed already. But, so far as I know, they have never been considered from the point of view which I adopt in this paper; and I believe that all the results which it contains are new.The majority of my formulae are 'elementary' in the technical sense of the word — they can (that is to say) be proved by a combination of processes involving only finite algebra and simple general theorems concerning infinite series 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: 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. For integers a and b, a ∣ b {displaystyle amid b} is read 'a divides b' and means that there is an integer c such that b = ac. Similarly, a ∤ b {displaystyle a mid b} is read 'a does not divide b'. The summation symbol means that d goes through all the positive divisors of m, e.g. ( a , b ) {displaystyle (a,,b)} is the greatest common divisor, ϕ ( n ) {displaystyle phi (n)} is Euler's totient function, μ ( n ) {displaystyle mu (n)} is the Möbius function, and ζ ( s ) {displaystyle zeta (s)} is the Riemann zeta function.