Ramanujan's sum

In number theory, Ramanujan's sum, usually denoted c_q, 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 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. c_q 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 = 1. Therefore, there are φ primitive q-th roots of unity.
Thus, the Ramanujan sum c_q 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 x^q − 1 = that
and this leads to the formula
published by Kluyver in 1906.
This shows that c_q is always an integer. Compare it with the formula

von Sterneck

It is easily shown from the definition that c_q 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 p^k 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 q the 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.

Ramanujan expansions

If is an arithmetic function, then a convergent infinite series of the form:
or of the form:
where the, is called a Ramanujan expansion of.
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 of prime numbers, namely that the series
converges to 0, and the results for and 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,,... 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 polynomial

σ_''k''(''n'')

is the divisor function., 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.
He proves that
and uses this to show that
Letting,
Note that the constant is the inverse 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.

Zero

For all,
This is equivalent to the prime number theorem.

''r''_2''s''(''n'') (sums of squares)

is the number of ways of representing as the sum of squares, counting different orders and signs as different
Ramanujan defines a function and references a paper 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
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,