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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 Moebius 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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