Search for a tool
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.

Results

Möbius Function -

Tag(s) : Arithmetics

Share
dCode and more

dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!
A suggestion ? a feedback ? a bug ? an idea ? Write to dCode!

Please, check our dCode Discord community for help requests!
NB: for encrypted messages, test our automatic cipher identifier!

Feedback and suggestions are welcome so that dCode offers the best 'Möbius Function' tool for free! Thank you!

# Möbius Function

## Mobius μ(N) Calculator

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

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

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

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)

## Source code

dCode retains ownership of the "Möbius Function" source code. Except explicit open source licence (indicated Creative Commons / free), the "Möbius Function" algorithm, the applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, breaker, translator), or the "Möbius Function" functions (calculate, convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) and all data download, script, or API access for "Möbius Function" are not public, same for offline use on PC, mobile, tablet, iPhone or Android app!
Reminder : dCode is free to use.

## Cite dCode

The copy-paste of the page "Möbius Function" or any of its results, is allowed (even for commercial purposes) as long as you credit dCode!
Exporting results as a .csv or .txt file is free by clicking on the export icon
Cite as source (bibliography):
Möbius Function on dCode.fr [online website], retrieved on 2024-07-18, https://www.dcode.fr/mobius-function

## Need Help ?

Please, check our dCode Discord community for help requests!
NB: for encrypted messages, test our automatic cipher identifier!