In number theory, the Liouville function, named after French mathematician Joseph Liouville and denoted λ ( n ) {\displaystyle \lambda (n)} , is an important arithmetic function. Its value is 1 {\displaystyle 1} if n {\displaystyle n} is the product of an even number of prime numbers, and − 1 {\displaystyle -1} if it is the product of an odd number of prime numbers.
Definition By the fundamental theorem of arithmetic, any positive integer n {\displaystyle n} can be represented uniquely as a product of powers of primes:
n = p 1 a 1 ⋯ p k a k {\displaystyle n=p_{1}^{a_{1}}\cdots p_{k}^{a_{k}}} , where p 1 , … , p k {\displaystyle p_{1},\dots ,p_{k}} are primes and the exponents a 1 , … , a k {\displaystyle a_{1},\dots ,a_{k}} are positive integers. The prime omega function Ω ( n ) {\displaystyle \Omega (n)} counts the number of primes in the factorization of n {\displaystyle n} with multiplicity:
Ω ( n ) = a 1 + a 2 + ⋯ + a k {\displaystyle \Omega (n)=a_{1}+a_{2}+\cdots +a_{k}} . Thus, the Liouville function is defined by
λ ( n ) = ( − 1 ) Ω ( n ) {\displaystyle \lambda (n)=(-1)^{\Omega (n)}}
(sequence A008836 in the OEIS).
Properties Since Ω ( n ) {\displaystyle \Omega (n)} is completely additive; i.e., Ω ( a b ) = Ω ( a ) + Ω ( b ) {\displaystyle \Omega (ab)=\Omega (a)+\Omega (b)} , then λ ( n ) {\displaystyle \lambda (n)} is completely multiplicative. Since 1 {\displaystyle 1} has no prime factors, Ω ( 1 ) = 0 {\displaystyle \Omega (1)=0} , so λ ( 1 ) = 1 {\displaystyle \lambda (1)=1} .
λ ( n ) {\displaystyle \lambda (n)} is also related to the Möbius function μ ( n ) {\displaystyle \mu (n)} : if we write n {\displaystyle n} as n = a 2 b {\displaystyle n=a^{2}b} , where b {\displaystyle b} is squarefree, then
λ ( n ) = μ ( b ) . {\displaystyle \lambda (n)=\mu (b).}
The sum of the Liouville function over the divisors of n {\displaystyle n} is the characteristic function of the squares:
∑ d | n λ ( d ) = { 1 if n is a perfect square, 0 otherwise. {\displaystyle \sum _{d|n}\lambda (d)={\begin{cases}1&{\text{if }}n{\text{ is a perfect square,}}\\0&{\text{otherwise.}}\end{cases}}}
Möbius inversion of this formula yields
λ ( n ) = ∑ d 2 | n μ ( n d 2 ) . {\displaystyle \lambda (n)=\sum _{d^{2}|n}\mu \left({\frac {n}{d^{2}}}\right).}
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