# LiouvilleLambda

gives the Liouville function .

# Details and Options

• LiouvilleLambda is also known as Liouville function.
• Integer mathematical function, suitable for both symbolic and numerical manipulation.
• gives 1 whenever the number of prime factors counting multiplicity of n is even and -1 otherwise.
• For a number n=u p1k1 pmkm with u a unit and pi primes, returns (-1)k1++km.
• With the setting , LiouvilleLambda is defined using factorization over Gaussian integers.
• LiouvilleLambda[m+In] automatically works over Gaussian integers.

# Examples

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## Basic Examples(2)

Compute the Liouville function at and :

Plot the LiouvilleLambda sequence for the first 20 numbers:

## Scope(8)

### Numerical Evaluation(4)

LiouvilleLambda works over integers:

Gaussian integers:

Compute for large integers:

### Symbolic Manipulation(4)

Reduce expressions:

Solve equations:

Equivalently:

## Options(1)

### GaussianIntegers(1)

Compute LiouvilleLambda over integers:

Gaussian integers:

## Applications(5)

### Basic Applications(2)

Highlight numbers for which Liouville's function is in blue and for which it is in red:

Histogram of the cumulative values of LiouvilleLambda:

### Number Theory(3)

LiouvilleLambda and MoebiusMu are related by the equation :

Use LiouvilleLambda to compute MoebiusMu:

Plot the summatory function :

## Properties & Relations(5)

LiouvilleLambda is a completely multiplicative function:

LiouvilleLambda gives when the argument is a product of an even number of primes:

Otherwise, it gives :

Use FactorInteger to compute LiouvilleLambda:

Use PrimeOmega to compute LiouvilleLambda:

gives for a perfect square n and otherwise:

## Neat Examples(3)

Plot LiouvilleLambda for the sum of two squares:

Plot the arguments of the Fourier transform of LiouvilleLambda:

Plot the Ulam spiral of LiouvilleLambda:

Wolfram Research (2008), LiouvilleLambda, Wolfram Language function, https://reference.wolfram.com/language/ref/LiouvilleLambda.html.

#### Text

Wolfram Research (2008), LiouvilleLambda, Wolfram Language function, https://reference.wolfram.com/language/ref/LiouvilleLambda.html.

#### CMS

Wolfram Language. 2008. "LiouvilleLambda." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/LiouvilleLambda.html.

#### APA

Wolfram Language. (2008). LiouvilleLambda. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/LiouvilleLambda.html

#### BibTeX

@misc{reference.wolfram_2023_liouvillelambda, author="Wolfram Research", title="{LiouvilleLambda}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/LiouvilleLambda.html}", note=[Accessed: 17-April-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2023_liouvillelambda, organization={Wolfram Research}, title={LiouvilleLambda}, year={2008}, url={https://reference.wolfram.com/language/ref/LiouvilleLambda.html}, note=[Accessed: 17-April-2024 ]}