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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

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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.

Answers to Questions

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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