In number theory, a branch of mathematics, Ramanujan's sum, usually denoted cq, is a function of two positive integer variables q and n defined by the formula: where = 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.
Notation
For integers a and b, is read "a divides b" and means that there is an integer c such that b = ac. Similarly, is read "a does not divide b". The summation symbol means that d goes through all the positive divisors of m, e.g. is the greatest common divisor, is Euler's totient function, is the Möbius function, and is the Riemann zeta function.
Formulas for ''c''''q''(''n'')
Trigonometry
These formulas come from the definition, Euler's formula and elementary trigonometric identities. and so on They show that cq is always real.
Kluyver
Let Then is a root of the equation. Each of its powers, is also a root. Therefore, since there are q of them, they are all of the roots. The numbers where 1 ≤ n ≤ q are called the q-th roots of unity. is called a primitiveq-th root of unity because the smallest value of n that makes is q. The other primitive q-th roots of unity are the numbers where = 1. Therefore, there are φ primitive q-th roots of unity. Thus, the Ramanujan sum cq is the sum of the n-th powers of the primitive q-th roots of unity. It is a fact that the powers of are precisely the primitive roots for all the divisors of q. Example. Let q = 12. 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 = that and this leads to the formula published by Kluyver in 1906. This shows that cq is always an integer. Compare it with the formula
von Sterneck
It is easily shown from the definition that cq is multiplicative when considered as a function of q for a fixed value of n: i.e. From the definition 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. The equivalence of it and Ramanujan's sum is due to Hölder.
Other properties of ''c''''q''(''n'')
For all positive integers q, For a fixed value of qthe absolute value of the sequence is bounded by φ, and for a fixed value of n the absolute value of the sequence is bounded by n. If q > 1 Let m1, m2 > 0, m = lcm. Then Ramanujan's sums satisfy an orthogonality property: Let n, k > 0. Then known as the Brauer - Rademacher identity. If n > 0 and a is any integer, we also have due to Cohen.
Table
Ramanujan expansions
If f is an arithmetic function, then a convergent infinite series of the form: or of the form: where the, is called a Ramanujan expansion of f. Ramanujan found expansions of some of the well-known functions of number theory. All of these results are proved in an "elementary" manner. The expansion of the zero function depends on a result from the analytic theory ofprime numbers, namely that the series converges to 0, and the results for r and r′ depend on theorems in an earlier paper. All the formulas in this section are from Ramanujan's 1918 paper.
Generating functions
The generating functions of the Ramanujan sums are Dirichlet series: is a generating function for the sequence cq, cq,... where q is kept constant, and is a generating function for the sequence c1, c2,... where n is kept constant. There is also the doubleDirichlet series
σ''k''(''n'')
σk is the divisor function. σ0, the number of divisors of n, is usually written d and σ1, the sum of the divisors of n, is usually written σ. If s > 0, Setting s = 1 gives If the Riemann hypothesis is true, and
''d''(''n'')
d = σ0 is the number of divisors of n, including 1 and n itself. where γ = 0.5772... is the Euler–Mascheroni constant.
''φ''(''n'')
φ 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 = φ is Euler's function. He proves that and uses this to show that Letting s = 1, Note that the constant is the inverse of the one in the formula for σ.
Λ(''n'')
unless n = pk is a power of a prime number, in which case it is the natural logarithm log p.
r2s is the number of way of representing n as the sum of 2ssquares, counting different orders and signs as different = 8, as 13 = 2 + 2 = 2 + Ramanujan defines a function δ2s and references a paper in which he proved that r2s = δ2s for s = 1, 2, 3, and 4. For s > 4 he shows that δ2s is a good approximation to r2s. s = 1 has a special formula: In the following formulas the signs repeat with a period of 4. and therefore,
(sums of triangles)
is the number of ways n can be represented as the sum of 2s triangular numbers The analysis here is similar to that for squares. Ramanujan refers to the same paper as he did for the squares, where he showed that there is a function such that for s = 1, 2, 3, and 4, and that for s > 4, is a good approximation to Again, s = 1 requires a special formula: If s is a multiple of 4, Therefore,
