# MoebiusMu

MoebiusMu[n]

gives the Möbius function .

# Details

• MoebiusMu is also known as Möbius function.
• Integer mathematical function, suitable for both symbolic and numerical manipulation.
• MoebiusMu[n] gives the sum of the primitive roots of unity.
• For a number n=u p1k1 pmkm with u a unit and pi primes, MoebiusMu[n]returns 0 unless all ki are equal to 1, in which case it gives (-1)m.
• MoebiusMu[m+In] automatically works over Gaussian integers.

# Examples

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## Basic Examples(2)

Compute the Möbius function at 11:

Plot the MoebiusMu sequence for the first 20 numbers:

## Scope(10)

### Numerical Evaluation(4)

MoebiusMu works over integers:

Gaussian integers:

Compute for large integers:

### Symbolic Manipulation(6)

Reduce expressions:

Solve equations:

Use MoebiusMu in a sum:

In a product:

Identify the MoebiusMu sequence:

Equivalently:

## Applications(11)

### Basic Applications(2)

Highlight numbers n for which in blue, numbers n for which in red and numbers n for which in black:

Histogram of the cumulative values of MoebiusMu:

The underlying distribution from the data:

### Number Theory(9)

Use MoebiusMu to test for a square-free number:

Use MoebiusMu to compute the number of terms in the Farey sequence:

Compare with the following:

Use MoebiusMu to compute MangoldtLambda:

gives 1 when and 0 when :

MoebiusMu is related to DivisorSigma through the Möbius inversion formulas:

MoebiusMu is related to PrimeNu through the following formula:

MoebiusMu satisfies the following identities:

Compute the number of polynomials over that are irreducible of degree n:

Irreducible polynomials modulo 5:

Distribution of irreducible polynomials modulo 5:

Logarithmic plot of the count for :

Plot :

## Properties & Relations(7)

MoebiusMu is a multiplicative function:

is 1 if n is a product of an even number of distinct primes:

is if it is a product of an odd number of primes:

is 0 if it has a multiple prime factor:

MoebiusMu is 0 for composite prime powers and for primes:

MoebiusMu is 0 for non-square-free integers:

Use PrimeNu to compute MoebiusMu for square-free numbers:

MoebiusMu is equal to the sum of the primitive roots of unity:

Compare with the following:

MoebiusMu can be expressed in terms of LiouvilleLambda and KroneckerDelta:

## Neat Examples(4)

Plot MoebiusMu for the sum of two squares:

Plot the arguments of the Fourier transform of MoebiusMu:

Plot the Ulam spiral of MoebiusMu:

Plot the values of that have :

Wolfram Research (1988), MoebiusMu, Wolfram Language function, https://reference.wolfram.com/language/ref/MoebiusMu.html.

#### Text

Wolfram Research (1988), MoebiusMu, Wolfram Language function, https://reference.wolfram.com/language/ref/MoebiusMu.html.

#### CMS

Wolfram Language. 1988. "MoebiusMu." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/MoebiusMu.html.

#### APA

Wolfram Language. (1988). MoebiusMu. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/MoebiusMu.html

#### BibTeX

@misc{reference.wolfram_2024_moebiusmu, author="Wolfram Research", title="{MoebiusMu}", year="1988", howpublished="\url{https://reference.wolfram.com/language/ref/MoebiusMu.html}", note=[Accessed: 24-July-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_moebiusmu, organization={Wolfram Research}, title={MoebiusMu}, year={1988}, url={https://reference.wolfram.com/language/ref/MoebiusMu.html}, note=[Accessed: 24-July-2024 ]}