Liouville function

Builds on Number of prime factors counted with multiplicity

Definition

The Liouville function $\lambda$ is the arithmetic function given by $\lambda(n) = (-1)^{\Omega(n)}$ for each positive integer $n$ . Here $\Omega(n)$ is the number of prime factors of n counted with multiplicity.

Remarks

For example, $\lambda(1) = 1$ , $\lambda(12) = -1$ and $\lambda(36) = 1$ . The function $\lambda$ is completely multiplicative, since $\Omega$ is completely additive. The sum $\sum_{d \mid n} \lambda(d)$ is $1$ if $n$ is a perfect square, and $0$ otherwise. On squarefree integers, $\lambda$ agrees with the Möbius function.

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Arithmetic function definition
Divisibility definition
Prime number definition
Prime factorization definition
Number of prime factors counted with multiplicity definition
Liouville function definition

Tag	`08RQAPXF1T`
URI	`m2://08RQAPXF1T`
Kind	definition
Writing	W1 stated
MSC	11A25
Defines	Liouville function
Uses	Number of prime factors counted with multiplicity

Liouville function

Definition

Remarks

Reading path

Further reading

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