In number theory, 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.
[1] 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.
is read "a divides b" and means that there is an integer c such that
The summation symbol means that d goes through all the positive divisors of m, e.g.
Then ζq is a root of the equation xq − 1 = 0.
ζq is called a primitive q-th root of unity because the smallest value of n that makes
The other primitive q-th roots of unity are the numbers
Therefore, there are φ(q) primitive q-th roots of unity.
Thus, the Ramanujan sum cq(n) is the sum of the n-th powers of the primitive q-th roots of unity.
It is a fact[3] that the powers of ζq are precisely the primitive roots for all the divisors of q.
Then Therefore, if is the sum of the n-th powers of all the roots, primitive and imprimitive, and by Möbius inversion, It follows from the identity xq − 1 = (x − 1)(xq−1 + xq−2 + ... + x + 1) that and this leads to the formula published by Kluyver in 1906.
Compare it with the formula It is easily shown from the definition that cq(n) is multiplicative when considered as a function of q for a fixed value of n:[5] i.e. From the definition (or Kluyver's formula) it is straightforward to prove that, if p is a prime number, and if pk is a prime power where k > 1, This result and the multiplicative property can be used to prove This is called von Sterneck's arithmetic function.
[6] The equivalence of it and Ramanujan's sum is due to Hölder.
Then[9] Ramanujan's sums satisfy an orthogonality property: Let n, k > 0.
If f (n) is an arithmetic function (i.e. a complex-valued function of the integers or natural numbers), then a convergent infinite series of the form: or of the form: where the ak ∈ C, is called a Ramanujan expansion[12] of f (n).
Ramanujan found expansions of some of the well-known functions of number theory.
All of these results are proved in an "elementary" manner (i.e. only using formal manipulations of series and the simplest results about convergence).
[13][14][15] The expansion of the zero function depends on a result from the analytic theory of prime numbers, namely that the series converges to 0, and the results for r(n) and r′(n) depend on theorems in an earlier paper.
[16] All the formulas in this section are from Ramanujan's 1918 paper.
The generating functions of the Ramanujan sums are Dirichlet series: is a generating function for the sequence cq(1), cq(2), ... where q is kept constant, and is a generating function for the sequence c1(n), c2(n), ... where n is kept constant.
There is also the double Dirichlet series The polynomial with Ramanujan sum's as coefficients can be expressed with cyclotomic polynomial[17] σk(n) is the divisor function (i.e. the sum of the k-th powers of the divisors of n, including 1 and n).
If s > 0, Setting s = 1 gives If the Riemann hypothesis is true, and
Euler's totient function φ(n) is the number of positive integers less than n and coprime to n. Ramanujan defines a generalization of it, if is the prime factorization of n, and s is a complex number, let so that φ1(n) = φ(n) is Euler's function.
[18] He proves that and uses this to show that Letting s = 1, Note that the constant is the inverse[19] of the one in the formula for σ(n).
Von Mangoldt's function Λ(n) = 0 unless n = pk is a power of a prime number, in which case it is the natural logarithm log p. For all n > 0, This is equivalent to the prime number theorem.
[20][21] r2s(n) is the number of ways of representing n as the sum of 2s squares, counting different orders and signs as different (e.g., r2(13) = 8, as 13 = (±2)2 + (±3)2 = (±3)2 + (±2)2.)
Ramanujan defines a function δ2s(n) and references a paper[22] in which he proved that r2s(n) = δ2s(n) for s = 1, 2, 3, and 4.
Ramanujan refers to the same paper as he did for the squares, where he showed that there is a function
Again, s = 1 requires a special formula: If s is a multiple of 4, Therefore, Let Then for s > 1, 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