## Möbius function and intervals

### Problem 464

The **Möbius function**, denoted `μ`(`n`), is defined as:

`μ`(`n`) = (-1)^{ω(n)}if`n`is squarefree (where`ω`(`n`) is the number of distinct prime factors of`n`)`μ`(`n`) = 0 if`n`is not squarefree.

Let P(`a`,`b`) be the number of integers `n` in the interval [`a`,`b`] such that `μ`(`n`) = 1.

Let N(`a`,`b`) be the number of integers `n` in the interval [`a`,`b`] such that `μ`(`n`) = -1.

For example, P(2,10) = 2 and N(2,10) = 4.

Let C(`n`) be the number of integer pairs (`a`,`b`) such that:

- 1 ≤
`a`≤`b`≤`n`, - 99·N(
`a`,`b`) ≤ 100·P(`a`,`b`), and - 99·P(
`a`,`b`) ≤ 100·N(`a`,`b`).

For example, C(10) = 13, C(500) = 16676 and C(10 000) = 20155319.

Find C(20 000 000).