Why it matters

Möbius inversion solves counting problems with divisor constraints. Powerful technique.

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The architecture

mu(n) = 0 if n has squared prime factor.

Otherwise: (-1)^k for k distinct primes.

Möbius function computationFactor nprimesCheck squaredif yes: 0Count primes(-1)^kPrecompute mu via sieve; Möbius inversion counts coprime pairs
Möbius steps.
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How it works end to end

Precompute via sieve.

Inversion: if f(n) = sum g(d) for d|n, then g(n) = sum mu(n/d) f(d).

Applications: counting coprime pairs.