Ramanujan's sum

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Not to be confused with Ramanujan summation.

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 introduced 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]

Notation[edit]

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]

Trigonometry[edit]

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.

Kluyver[edit]

Let Then ζq is a root of the equation xq − 1 = 0. Each of its powers ζq, ζq2, ... ζqq = ζq0 = 1 is also a root. Therefore, since there are q of them, they are all of the roots. The numbers ζqn 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 ζqn = 1 is q. The other primitive q-th roots of are the numbers ζqa 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

ζ12, ζ125, ζ127, and ζ1211 are the primitive twelfth roots of unity,
ζ122 and ζ1210 are the primitive sixth roots of unity,
ζ123 = i and ζ129 = −i are the primitive fourth roots of unity,
ζ124 and ζ128 are the primitive third roots of unity,
ζ126 = −1 is the primitive second root of unity, and
ζ1212 = 1 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 Sterneck[edit]

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

cq(1), cq(2), ... is bounded by φ(q), and

for a fixed value of n the absolute value of the sequence

c1(n), c2(n), ... 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.

Table[edit]

1+1+1+1+1 1+1+1+1+1 1+1+1+1+1 1+1+1 +(1×0)

Ramanujan Sum cs(n)
  n
5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 276 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[edit]

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 functions[edit]

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,

and

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.

Zero[edit]

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.

If s ≡ 0 (mod 4),

If s ≡ 2 (mod 4),

If s ≡ 1 (mod 4) and s > 1,

If s ≡ 3 (mod 4),

and therefore,

r2s(n) (sums of triangles)[edit]

r2s(n) 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 δ′2s(n) such that r2s(n) = δ′2s(n) for s = 1, 2, 3, and 4, and that for s > 4, δ′2s(n) is a good approximation to r2s(n).

Again, s = 1 requires a special formula:

If s is a multiple of 4,

If s is twice an odd number,

If s is an odd number and s > 1,

Therefore,

Sums[edit]

Let

Then for s > 1,

See also[edit]

Notes[edit]

  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

References[edit]

  • 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". Canad. 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 links[edit]

  • Tóth, László (2011). "Sums of products of Ramanujan sums". arXiv:1104.1906Freely accessible.