Liouville Function

The Liouville function, denoted by λ(n) and named after Joseph Liouville, is an important function in number theory.

If n is a positive integer, then λ(n) is defined as:

where Ω(n) is the number of prime factors of n, counted with multiplicity (sequence A008836 in OEIS).

λ is completely multiplicative since Ω(n) is completely additive. The number one has no prime factors, so Ω(1) = 0 and therefore λ(1) = 1. The Liouville function satisfies the identity:


\sum_{d|n}\lambda(d) =
\begin{cases}
1 & \text{if }n\text{ is a perfect square,} \\
0 & \text{otherwise.}
\end{cases}

The Liouville function's Dirichlet inverse is the absolute value of the Mobius function.

Read more about Liouville Function:  Series, Conjectures

Famous quotes containing the word function:

    The information links are like nerves that pervade and help to animate the human organism. The sensors and monitors are analogous to the human senses that put us in touch with the world. Data bases correspond to memory; the information processors perform the function of human reasoning and comprehension. Once the postmodern infrastructure is reasonably integrated, it will greatly exceed human intelligence in reach, acuity, capacity, and precision.
    Albert Borgman, U.S. educator, author. Crossing the Postmodern Divide, ch. 4, University of Chicago Press (1992)