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# 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 = p_{1}

^{a1}⋅p

_{2}

^{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)

^{k}Π (p

_{j})

^{x}

Σ μ(d)φ(d) = (-1)

^{k}Π (p

_{j}- 2)

### Infinite Sums Involving μ(n)

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