Let and be positive integers. Define
is called Ramanujan’s sum.
2 Fourier transform on and the principal Dirichlet character modulo
For , the Fourier transform of is defined by
Define by if and if . is called the principal Dirichlet character modulo . The Fourier transform of is
Therefore we can write Ramanujan’s sum as , thus .
The above gives us an expression for as a multiple of the Fourier transform of the principal Dirichlet character modulo . , and we can write the Fourier transform of as
3 Dirichlet series
Here I am following Titchmarsh in §1.5 of his The theory of the Riemann zeta-function, second ed. Let be the Möbius function. The Möbius inversion formula states that if
( is a sum over the positive divisors of .)
We have (this is not supposed to be obvious)
Therefore by the Möbius inversion formula we have
(Hence , where .)
If then , and if then . (To show the second statement: multiply the sum by , and check that this product is equal to the original sum. Since we multplied the sum by a number that is not , the sum must be equal to .) Thus we can express the Möbius function using Ramanujan’s sum as .
Because if and if , we have
here we used that
On the other hand, if rather than sum over we sum over , then we obtain