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.

Möbius Function - dCode

Tag(s) : Arithmetics

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The function $ μ(n) $, called the Möbius function (or Moebius), 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

Automatic method: indicate the value $ n $ for which to calculate $ μ(n) $ in dCode (above)

Manual method: 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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*Möbius Function* on dCode.fr [online website], retrieved on 2022-09-30,

mobius,moebius,mu,prime,number

https://www.dcode.fr/mobius-function

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