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Möbius Function

Tool to calculate the value of the function μ (Mu) of Möbius (or Moebius) which has a value of -1, 0 or 1 according to its prime numbers decomposition.

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Möbius Function -

Tag(s) : Arithmetics

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# Möbius Function

## Mobius µ(N) Calculator

Tool to calculate the value of the function μ (Mu) of Möbius (or Moebius) which has a value of -1, 0 or 1 according to its prime numbers decomposition.

### What is the Mobius Mu function? (Definition)

The function $μ(n)$, called the Möbius function, is defined for any integer $n> 0$ of the set $\mathbb{N}*$ in the set of 3 values $\{-1, 0, 1 \}$.

$μ(n)$ is $0$ if $n$ has for divisor a perfect square (other than 1)

$μ(n)$ is $1$ if $n$ has for divisors an even number of prime numbers

$μ(n)$ is $-1$ if $n$ has for divisors an odd number of prime numbers

### How to calculate the value of the Mobius function?

The image of $μ(n)$ depends on the prime number decomposition of $n$. If a prime number appears several times in the decomposition, then $μ(n) = 0$, otherwise, if the decomposition has an even number of prime numbers, then $μ(n) = 1$ and otherwise with an odd number of prime numbers $μ(n) = -1$.

Example: $12 = 2 \times 2 \times 3$ so $μ(12) = 0$ because $2$ appears twice, and so $12$ is divisible by $4$, a perfect square

Example: $1234 = 2 \times 617$ therefore $μ(12) = 1$ because the decomposition has 2 distinct primes (2 is an even number)

Example: $12345 = 3 \times 5 \times 823$ so $μ(12) = -1$ because the decomposition has 3 distinct prime numbers (3 is an odd number)

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