Möbius Mu Function Calculator

In elementary number theory, the Möbius Mu function, denoted μ(n), returns a value of -1, 0, or 1 depending on the factorization of n.

If n is greater than 1 and divisible by a square integer, μ(n) = 0. Otherwise, if n = 1 or n has an even number of distinct prime factors, μ(n) = 1. And if n has an odd number of distinct prime factors, μ(n) = -1.

Like the most commonly studied number theoretic functions, μ(n) is multiplicative. That is, if GCD(m,n) = 1, then μ(m)μ(n) = μ(mn).

The Möbius μ function by itself is not so interesting, however, it plays an important role in many number theoretic identities and expressions.

The Möbius Inversion Formula

One application of μ(n) is in the Möbius Inversion Formula. If F(n) and f(n) are number theoretic functions related by the expression

F(n) = Σ f(d)

where d ranges over the divisors of n, then f(n) can be written as

f(n) = Σ μ(d)F(n/d)

or equivalently

f(n) = Σ μ(n/d)F(d)

For example, consider the following summation formulas for the divisor function and Euler's Totient function respectively:

σₓ(n) = Σ d^x

n = Σ φ(d)

The Möbius Inversion Formula gives us two new expressions:

n^x = Σ μ(d)σₓ(n/d)

φ(n) = Σ μ(n/d)d

More Properties of μ(n)

Consider another identity: Suppose the prime factorization of n is n = p₁^a1⋅p₂^a2⋅...⋅p_k^ak. If f(n) is any multiplicative function, then

Σ μ(d)f(d) = Π (1 - f(p_j))

where the p_j's are the prime factors of n, and j ranges from 1 to k (the number of distinct prime factors of n). Some examples of this second identity are

Σ μ(d)σₓ(d) = (-1)^kΠ (p_j)^x
Σ μ(d)φ(d) = (-1)^kΠ (p_j - 2)

Infinite Sums Involving μ(n)

The Riemann Zeta function ζ(s) is defined by the equation

ζ(s) = Σ 1/n^s.

for n = 1 to infinity. If the numerator 1 is replaced with μ(n), then

1/ζ(s) = Σ μ(n)/n^s.

If you set s = 1 and s = 2, you obtain the infinite sums

Σ μ(n)/n = 0, and

Σ μ(n)/n² = 6/π²

Two more remarkable sums are

Σ μ(n)ln(n)/n = -1

Σ [μ(n)/n]² = 15/π²

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