PlanetMath: Liouville function

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Liouville function

(Definition)

The Liouville function is defined by $\lambda (1) = 1$ and $\lambda (n) = (-1)^{k_1 + k_2 + \cdots + k_r}$ , if the prime factorization of is $n = p_1^{k_1} p_2^{k_2} \cdots p_r^{k_r}$ (where each is positive). This function is completely multiplicative and satisfies the identity

$\begin{displaymath} \sum_{d\vert n} \lambda (d) = \begin{cases} 1 &\text{if $n=m^2$\ for some integer $m$}\ 0 &\text{otherwise,} \end{cases}\end{displaymath}$

where the sum runs over all positive divisors of

"Liouville function" is owned by KimJ.

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Keywords:

number theory

Attachments:: table of values of the Liouville function and its summatory function (Data Structure) by PrimeFan

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Cross-references: divisors, sum, completely multiplicative, function, positive, prime factorization
There are 7 references to this entry.

This is version 7 of Liouville function, born on 2001-10-16, modified 2007-04-15.
Object id is 257, canonical name is LiouvilleFunction.
Accessed 2722 times total.

Classification:

AMS MSC:

11A25 (Number theory :: Elementary number theory :: Arithmetic functions; related numbers; inversion formulas)

Pending Errata and Addenda

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