Möbius Function Calculator

What Is the Möbius Function? The Möbius function μ(n) is a fundamental multiplicative function in number theory, defined as: μ(1) = 1. μ(n) = (−1)^k if n is the product of k distinct primes (squarefree). μ(n) = 0 if n has any squared prime factor (i.e., p² | n for some prime p). Named after August Ferdinand Möbius, German mathematician, 1832.

The Three Cases μ(n) = 1: n is squarefree with an even number of distinct prime factors. Examples: 1 (zero primes, and zero is even), 6 = 2×3, 10 = 2×5, 210 = 2×3×5×7. μ(n) = −1: n is squarefree with an odd number of distinct prime factors. Examples: 2, 3, 5 and every other prime (one factor each), 30 = 2×3×5, 105 = 3×5×7. μ(n) = 0: n is not squarefree, meaning some prime appears more than once. Examples: 4 (2²), 8 (2³), 9 (3²), 12 (2²×3), 18 (2×3²).

Count distinct primes, not repeats, and count them carefully. 30 is a common slip: it looks like a small friendly number but it carries three primes, so μ(30) = −1, not +1. Only the parity of that count matters, never the size of the primes involved.

Squarefreeness A number n is squarefree if no prime squared divides it. Squarefree: 1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, … Non-squarefree: 4, 8, 9, 12, 16, 18, 20, 24, 25, … About 6/π² ≈ 60.8% of all positive integers are squarefree.

Key Properties Multiplicativity: if gcd(m,n) = 1, then μ(mn) = μ(m)μ(n). Sum over divisors: Σ_{d|n} μ(d) = 1 if n=1, else 0. This is used to prove many results in analytic number theory. Dirichlet series: Σ μ(n)/nˢ = 1/ζ(s), where ζ is the Riemann zeta function. This inverse relationship makes μ(n) central to the study of the zeta function.

Möbius Inversion Formula If f(n) = Σ_{d|n} g(d), then g(n) = Σ_{d|n} μ(d) f(n/d). This lets you “invert” sums over divisors, and it is used constantly in analytic number theory. Example: Euler’s totient φ(n) = Σ_{d|n} μ(d) × (n/d).

Connection to the Riemann Hypothesis The Mertens function M(x) = Σ_{n≤x} μ(n). The Riemann Hypothesis is equivalent to: M(x) = O(x^(1/2+ε)) for every ε > 0. In other words, the Möbius function “averages out” quickly if and only if the Riemann Hypothesis is true. The Mertens conjecture (|M(x)| < √x) was disproved in 1985, though the counterexamples are astronomically large.

Applications Counting primitive Pythagorean triples, squarefree numbers, and primitive roots. Inclusion-exclusion in combinatorics via the Möbius inversion formula. Signal processing: Möbius function appears in the study of cyclotomic polynomials.