Ramanujan's sum

Ramanujan's sum
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. 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 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)
Trigonometry
These formulas come from the definition, Euler's formula and elementary trigonometric identities.
and so on (.. ...). cq(n) is always an integer.
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 fact 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.
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: 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. The equivalence of it and Ramanujan's sum is due to Hölder.
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). 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(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 , is called a Ramanujan expansion 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).
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.
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(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)
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)
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)
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.
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(n).
Λ(n)
Von Mangoldt's function unless n = pk is a power of a prime number, in which case it is the natural logarithm log p.
Zero
For all n > 0,
This is equivalent to the prime number theorem.
r2s(n) (sums of squares)
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 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,
r′2s(n) (sums of triangles)
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,
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
Number theory
Squares in number theory
Srinivasa Ramanujan