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.Ramanujan, On Certain Trigonometric Sums ... 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.(Papers, p. 179). In a footnote cites pp. 360–370 of the Dirichlet–Dedekind , 4th ed. 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.Nathanson, ch. 8.
cq(n) is always an integer.
Notation
For integers a and b, is read "a divides b" and means that there exists an integer c such that ; 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. is the Riemann zeta function.
Formulas for cq(n)
Trigonometry The following formulas come from the definition, Euler's formula and elementary trigonometric identities:
and so on (, , , ,.., ,...).
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 primitive q-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 (a, q) = 1. 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 factHardy & Wright, Thms 65, 66 that the powers of are precisely the primitive roots for all the divisors of q.
Example. Let q = 12. Then
and are the primitive twelfth roots of unity,
and are the primitive sixth roots of unity,
and are the primitive fourth roots of unity,
and are the primitive third roots of unity,
is the primitive second root of unity, and
is the primitive first root of unity.
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.G. H. Hardy, P. V. Seshu Aiyar, & B. M. Wilson, notes to On certain trigonometrical sums ..., Ramanujan, Papers, p. 343
This shows that cq(n) is always an integer. Compare it with the formula
von Sterneck 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:Schwarz & Spilken (1994) p.16 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.B. Berndt, commentary to On certain trigonometrical sums..., Ramanujan, Papers, p. 371 The equivalence of it and Ramanujan's sum is due to Hölder.Knopfmacher, p. 196Hardy & Wright, p. 243
Other properties of cq(n) For all positive integers q,
For a fixed value of q the absolute value of the sequence is bounded by φ(q), 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(m1, m2). ThenTóth, external links, eq. 6 Ramanujan's sums satisfy an orthogonality property:
Let n, k > 0. ThenTóth, external links, eq. 17.
known as the Brauer–Rademacher identity.
If n > 0 and a is any integer, we also haveTóth, external links, eq. 8.
due to Cohen.
Table
+ Ramanujan sum cs(n) n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 s 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 −1 1 −1 1 −1 1 −1 1 −1 1 −1 1 −1 1 −1 1 −1 1 −1 1 −1 1 −1 1 −1 1 −1 1 −1 1 3 −1 −1 2 −1 −1 2 −1 −1 2 −1 −1 2 −1 −1 2 −1 −1 2 −1 −1 2 −1 −1 2 −1 −1 2 −1 −1 2 4 0 −2 0 2 0 −2 0 2 0 −2 0 2 0 −2 0 2 0 −2 0 2 0 −2 0 2 0 −2 0 2 0 −2 5 −1 −1 −1 −1 4 −1 −1 −1 −1 4 −1 −1 −1 −1 4 −1 −1 −1 −1 4 −1 −1 −1 −1 4 −1 −1 −1 −1 4 6 1 −1 −2 −1 1 2 1 −1 −2 −1 1 2 1 −1 −2 −1 1 2 1 −1 −2 −1 1 2 1 −1 −2 −1 1 2 7 −1 −1 −1 −1 −1 −1 6 −1 −1 −1 −1 −1 −1 6 −1 −1 −1 −1 −1 −1 6 −1 −1 −1 −1 −1 −1 6 −1 −1 8 0 0 0 −4 0 0 0 4 0 0 0 −4 0 0 0 4 0 0 0 −4 0 0 0 4 0 0 0 −4 0 0 9 0 0 −3 0 0 −3 0 0 6 0 0 −3 0 0 −3 0 0 6 0 0 −3 0 0 −3 0 0 6 0 0 −3 10 1 −1 1 −1 −4 −1 1 −1 1 4 1 −1 1 −1 −4 −1 1 −1 1 4 1 −1 1 −1 −4 −1 1 −1 1 4 11 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 10 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 10 −1 −1 −1 −1 −1 −1 −1 −1 12 0 2 0 −2 0 −4 0 −2 0 2 0 4 0 2 0 −2 0 −4 0 −2 0 2 0 4 0 2 0 −2 0 −4 13 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 12 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 12 −1 −1 −1 −1 14 1 −1 1 −1 1 −1 −6 −1 1 −1 1 −1 1 6 1 −1 1 −1 1 −1 −6 −1 1 −1 1 −1 1 6 1 −1 15 1 1 −2 1 −4 −2 1 1 −2 −4 1 −2 1 1 8 1 1 −2 1 −4 −2 1 1 −2 −4 1 −2 1 1 8 16 0 0 0 0 0 0 0 −8 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 −8 0 0 0 0 0 0 17 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 16 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 18 0 0 3 0 0 −3 0 0 −6 0 0 −3 0 0 3 0 0 6 0 0 3 0 0 −3 0 0 −6 0 0 −3 19 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 18 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 20 0 2 0 −2 0 2 0 −2 0 −8 0 −2 0 2 0 −2 0 2 0 8 0 2 0 −2 0 2 0 −2 0 −8 21 1 1 −2 1 1 −2 −6 1 −2 1 1 −2 1 −6 −2 1 1 −2 1 1 12 1 1 −2 1 1 −2 −6 1 −2 22 1 −1 1 −1 1 −1 1 −1 1 −1 −10 −1 1 −1 1 −1 1 −1 1 −1 1 10 1 −1 1 −1 1 −1 1 −1 23 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 22 −1 −1 −1 −1 −1 −1 −1 24 0 0 0 4 0 0 0 −4 0 0 0 −8 0 0 0 −4 0 0 0 4 0 0 0 8 0 0 0 4 0 0 25 0 0 0 0 −5 0 0 0 0 −5 0 0 0 0 −5 0 0 0 0 −5 0 0 0 0 20 0 0 0 0 −5 26 1 −1 1 −1 1 −1 1 −1 1 −1 1 −1 −12 −1 1 −1 1 −1 1 −1 1 −1 1 −1 1 12 1 −1 1 −1 27 0 0 0 0 0 0 0 0 −9 0 0 0 0 0 0 0 0 −9 0 0 0 0 0 0 0 0 18 0 0 0 28 0 2 0 −2 0 2 0 −2 0 2 0 −2 0 −12 0 −2 0 2 0 −2 0 2 0 −2 0 2 0 12 0 2 29 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 28 −1 30 −1 1 2 1 4 −2 −1 1 2 −4 −1 −2 −1 1 −8 1 −1 −2 −1 −4 2 1 −1 −2 4 1 2 1 −1 8
Ramanujan expansions If 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 , is called a Ramanujan expansionB. Berndt, commentary to On certain trigonometrical sums..., Ramanujan, Papers, pp. 369–371 of .
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).Ramanujan, On certain trigonometrical sums... 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(Papers, p. 179)The theory of formal Dirichlet series is discussed in Hardy & Wright, § 17.6 and in Knopfmacher.Knopfmacher, ch. 7, discusses Ramanujan expansions as a type of Fourier expansion in an inner product space which has the as an orthogonal basis.
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 and depend on theorems in an earlier paper.Ramanujan, On Certain Arithmetical Functions
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 , , ... where is kept constant, and
is a generating function for the sequence , , ... where is kept constant.
There is also the double Dirichlet series
The polynomial with Ramanujan sum's as coefficients can be expressed with cyclotomic polynomialNicol, p. 1
σk(n) is the divisor function (i.e. the sum of the -th powers of the divisors of , including 1 and ). , the number of divisors of , is usually written and , the sum of the divisors of , is usually written .
If ,
Setting gives
If the Riemann hypothesis is true, and
d(n) is the number of divisors of , including 1 and itself.
where is the Euler–Mascheroni constant.
φ(n) Euler's totient function is the number of positive integers less than and coprime to . Ramanujan defines a generalization of it, if
is the prime factorization of , and is a complex number, let
so that is Euler's function.This is Jordan's totient function, .
He proves that
and uses this to show that
Letting ,
Note that the constant is the inverseCf. Hardy & Wright, Thm. 329, which states that of the one in the formula for .
Λ(n) Von Mangoldt's function unless is a power of a prime number, in which case it is the natural logarithm . For
Zero For all ,
This is equivalent to the prime number theorem.Hardy, Ramanujan, p. 141B. Berndt, commentary to On certain trigonometrical sums..., Ramanujan, Papers, p. 371
r2s(n) (sums of squares) is the number of ways of representing as the sum of squares, counting different orders and signs as different (e.g., , as .)
Ramanujan defines a function and references a paperRamanujan, On Certain Arithmetical Functions in which he proved that for . For he shows that is a good approximation to .
has a special formula:
In the following formulas the signs repeat with a period of 4.
and therefore,
r′2s(n) (sums of triangles) is the number of ways can be represented as the sum of triangular numbers (i.e. the numbers 1, 3 = 1 + 2, 6 = 1 + 2 + 3, 10 = 1 + 2 + 3 + 4, 15, ...; the -th triangular number is given by the formula .)
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 , and that for , is a good approximation to
Again, requires a special formula:
If is a multiple of 4,
Therefore,
Sums Let
Then for ,
See also Gaussian period Kloosterman sum
Notes
References
.
(pp. 179–199 of his Collected Papers)
(pp. 136–163 of his Collected Papers)
External links