Liouville's function lambda(n)

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The Liouville lambda function $\lambda (n)=(-1)^{\Omega (n)}$, where $\Omega (n)$ is the number of prime factors function (sometimes called "big omega"), tells whether $n$ has an odd or even number of prime factors, not necessarily distinct. Thus $\lambda (n)=-1$ when $n$ has an odd number of prime factors, and $\lambda (n)=1$ if $n$ has an even number of factors.

For example, $\lambda (26)=1$ because 26 has two prime factors (2 and 13) and so $\lambda (26)=(-1)^{2}=1$. $\lambda (27)=-1$ because 27 has three prime factors (3 thrice) and so $\lambda (27)=(-1)^{3}=-1$.

The function is completely multiplicative, meaning that $\lambda (mn)=\lambda (m)\lambda (n)$ regardless of whether $\gcd(m,n)=1$ or not. With $\lambda (28)=-1$ we exhibit that the function is completely multiplicative, since $\lambda (28)=\lambda (2)\lambda (14)=\lambda (4)\lambda (7)=-1\times 1=1$. See A008836 for more values.

The Liouville lambda function is sometimes notated $\psi (n)$.

$\scriptstyle \lambda (n),\,1\,\leq \,n\,\leq \,257,\,$ where $\scriptstyle \lambda (n)\,$ is Liouville's function lambda(n) (A008836)

1     $\lambda (n)\,$
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−1

Properties

Completely multiplicative

$\scriptstyle \lambda (n)\,$ is a completely multiplicative arithmetic function, i.e.

$\lambda (mn)=\lambda (m)\cdot \lambda (n),\ m\,\geq \,1,\ n\,\geq \,1.\,$

Theorem. The Liouville lambda function is completely multiplicative. Given $\{m,n\}\in \mathbb {Z} ^{+}$, the equality $\lambda (mn)=\lambda (m)\lambda (n)$ holds even when $\gcd(m,n)>1$.

Proof. Remember that $\Omega (mn)=\Omega (m)+\Omega (n)$ whether $\gcd(m,n)=1$ or not (whereas with the number of distinct prime factors function this would not be the case). Therefore, if $\Omega (m)$ is even and $\Omega (n)$ is also even, then so is $\Omega (mn)$ and thus $\lambda (mn)=\lambda (m)\lambda (n)=(-1)^{\Omega (m)}(-1)^{\Omega (n)}=1\times 1=1$. If $\Omega (m)$ is even but $\Omega (n)$ is odd, then $\Omega (mn)$ is odd and thus $\lambda (mn)=1\times -1=1$. But if both $\Omega (m)$ and $\Omega (n)$ are odd, then $\Omega (mn)$ is even and thus $\lambda (mn)=-1\times -1=1$. ENDOFPROOFMARK

Sum over the divisors $d\,$ of $n\,$ of $\lambda (d)\,$

$\sum _{d|n}\lambda (d)={\begin{cases}1&{\mbox{if }}n{\mbox{ is a square,}}\;\\0&{\mbox{otherwise.}}\;\end{cases}}\,$

Dirichlet generating function

The Dirichlet generating function of $\scriptstyle \lambda (n)\,$ is

$D_{\{\lambda (n)\}}(s)\equiv \sum _{n=1}^{\infty }{\frac {\lambda (n)}{n^{s}}}={\frac {\zeta (2s)}{\zeta (s)}},\ s\,>\,1,\,$

where $\scriptstyle \zeta (s)\,$ is the Riemann zeta function.

Liouville's summatory function L(n)

As the Mertens function is to the Möbius function, so is the Liouville summatory function $L(n)=\sum _{i=1}^{n}\lambda (i)$ (see A002819). See Pólya's conjecture.

The partial sums of Liouville's function $\scriptstyle \lambda (n)\,$ give Liouville's summatory function L(n)

$L(n)\equiv \sum _{k=1}^{n}\lambda (k)=\sum _{k=1}^{n}(-1)^{\Omega (k)}\,$

Sequences

Liouville's function $\scriptstyle \lambda (n),\ n\geq 1$ (A008836) gives the sequence

{1, −1, −1, 1, −1, 1, −1, −1, 1, 1, −1, −1, −1, 1, 1, 1, −1, −1, −1, −1, 1, 1, −1, 1, 1, 1, −1, −1, −1, −1, −1, −1, 1, 1, 1, 1, −1, 1, 1, 1, −1, −1, −1, −1, −1, 1, −1, −1, 1, −1, 1, −1, −1, 1, 1, 1, 1, 1, ...}

The partial sums $\scriptstyle L(n)\equiv \sum _{i=1}^{n}\lambda (i),\ n\geq 1$ (A002819) give the sequence

{1, 0, −1, 0, −1, 0, −1, −2, −1, 0, −1, −2, −3, −2, −1, 0, −1, −2, −3, −4, −3, −2, −3, −2, −1, 0, −1, −2, −3, −4, −5, −6, −5, −4, −3, −2, −3, −2, −1, 0, −1, −2, −3, −4, −5, −4, −5, −6, −5, −6, −5, −6, −7, −6, −5, ...}

See also

• A026424 Number of prime divisors (counted with multiplicity) is odd; Liouville function lambda(n) (A008836) is negative.
• A028260 Numbers n such that number of prime divisors of n (counted with multiplicity) is even; Liouville function lambda(n) (A008836) is positive.

• A056912 Odd squarefree numbers for which the number of prime divisors is odd.
• A056913 Odd squarefree numbers for which the number of prime divisors is even.

• Liouville's summatory function L(n)
• A002819 Liouville's function L(n) = partial sums of A008836.

• Omega(n), total number of primes dividing n