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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.[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.[2]

Contents

NotationEdit

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 cq(n)Edit

TrigonometryEdit

These formulas come from the definition, Euler's formula   and elementary trigonometric identities.

 

and so on (OEISA000012, OEISA033999, OEISA099837, OEISA176742,.., OEISA100051,...) They show that cq(n) is always real.

KluyverEdit

Let   Then ζq is a root of the equation xq − 1 = 0. 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 ≤ nq are called the q-th roots of unity. ζq 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 fact[3] that the powers of ζq 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.[4]

This shows that cq(n) is always an integer. Compare it with the formula

 

von SterneckEdit

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.[7][8]

Other properties of cq(n)Edit

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). Then[9] Ramanujan's sums satisfy an orthogonality property:

 

Let n, k > 0. Then[10]

 

known as the Brauer - Rademacher identity.

If n > 0 and a is any integer, we also have[11]

 

due to Cohen.

TableEdit

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 expansionsEdit

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 akC, 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.

Generating functionsEdit

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

 

σk(n)Edit

σk(n) is the divisor function (i.e. the sum of the k-th powers of the divisors of n, including 1 and n). σ0(n), the number of divisors of n, is usually written d(n) and σ1(n), the sum of the divisors of n, is usually written σ(n).

If s > 0,

 

Setting s = 1 gives

 

If the Riemann hypothesis is true, and  

 

d(n)Edit

d(n) = σ0(n) is the number of divisors of n, including 1 and n itself.

 

where γ = 0.5772... is the Euler–Mascheroni constant.

φ(n)Edit

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.[17]

He proves that

 

and uses this to show that

 

Letting s = 1,

 

Note that the constant is the inverse[18] of the one in the formula for σ(n).

Λ(n)Edit

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.

 

ZeroEdit

For all n > 0,

 

This is equivalent to the prime number theorem.[19][20]

r2s(n) (sums of squares)Edit

r2s(n) is the number of way 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[21] in which he proved that r2s(n) = δ2s(n) for s = 1, 2, 3, and 4. For s > 4 he shows that δ2s(n) is a good approximation to r2s(n).

s = 1 has a special formula:

 

In the following formulas the signs repeat with a period of 4.

 

and therefore,

 

  (sums of triangles)Edit

  is the number of ways n can be represented as the sum of 2s triangular numbers (i.e. the numbers 1, 3 = 1 + 2, 6 = 1 + 2 + 3, 10 = 1 + 2 + 3 + 4, 15, ...; the n-th triangular number is given by the formula n(n + 1)/2.)

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,

 

SumsEdit

Let

 

Then for s > 1,

 

See alsoEdit

NotesEdit

  1. ^ 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 Vorlesungen über Zahlentheorie, 4th ed.
  2. ^ Nathanson, ch. 8
  3. ^ Hardy & Wright, Thms 65, 66
  4. ^ G. H. Hardy, P. V. Seshu Aiyar, & B. M. Wilson, notes to On certain trigonometrical sums ..., Ramanujan, Papers, p. 343
  5. ^ Schwarz & Spilken (1994) p.16
  6. ^ B. Berndt, commentary to On certain trigonometrical sums..., Ramanujan, Papers, p. 371
  7. ^ Knopfmacher, p. 196
  8. ^ Hardy & Wright, p. 243
  9. ^ Tóth, external links, eq. 6
  10. ^ Tóth, external links, eq. 17.
  11. ^ Tóth, external links, eq. 8.
  12. ^ B. Berndt, commentary to On certain trigonometrical sums..., Ramanujan, Papers, pp. 369–371
  13. ^ 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)
  14. ^ The theory of formal Dirichlet series is discussed in Hardy & Wright, § 17.6 and in Knopfmacher.
  15. ^ Knopfmacher, ch. 7, discusses Ramanujan expansions as a type of Fourier expansion in an inner product space which has the cq as an orthogonal basis.
  16. ^ Ramanujan, On Certain Arithmetical Functions
  17. ^ This is Jordan's totient function, Js(n).
  18. ^ Cf. Hardy & Wright, Thm. 329, which states that  
  19. ^ Hardy, Ramanujan, p. 141
  20. ^ B. Berndt, commentary to On certain trigonometrical sums..., Ramanujan, Papers, p. 371
  21. ^ Ramanujan, On Certain Arithmetical Functions

ReferencesEdit

  • Hardy, G. H. (1999), Ramanujan: Twelve Lectures on Subjects Suggested by his Life and Work, Providence RI: AMS / Chelsea, ISBN 978-0-8218-2023-0
  • Nathanson, Melvyn B. (1996), Additive Number Theory: the Classical Bases, Graduate Texts in Mathematics, 164, Springer-Verlag, Section A.7, ISBN 0-387-94656-X, Zbl 0859.11002.
  • Nicol, C. A. (1962). "Some formulas involving Ramanujan sums". Can. J. Math. 14: 284–286. doi:10.4153/CJM-1962-019-8.
  • Ramanujan, Srinivasa (1918), "On Certain Trigonometric Sums and their Applications in the Theory of Numbers", Transactions of the Cambridge Philosophical Society, 22 (15): 259–276 (pp. 179–199 of his Collected Papers)
  • Ramanujan, Srinivasa (1916), "On Certain Arithmetical Functions", Transactions of the Cambridge Philosophical Society, 22 (9): 159–184 (pp. 136–163 of his Collected Papers)
  • Schwarz, Wolfgang; Spilker, Jürgen (1994), Arithmetical Functions. An introduction to elementary and analytic properties of arithmetic functions and to some of their almost-periodic properties, London Mathematical Society Lecture Note Series, 184, Cambridge University Press, ISBN 0-521-42725-8, Zbl 0807.11001

External linksEdit