Möbius function and intervals

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).

Written by gamwe6 who lives and works in San Francisco building useful things. You should follow him on Twitter