Liouville function \(\lambda(n)\) refresher

The Liouville function is a simple, oscillatory completely multiplicative function built from \(\Omega(n)\) (the total number of prime factors with multiplicity). See [Tenenbaum, 2015].

Definition

Define

\[ \lambda(n)=(-1)^{\Omega(n)}. \]

So \(\lambda(n)=1\) if \(n\) has an even total number of prime factors (with multiplicity), and \(\lambda(n)=-1\) otherwise.

Examples:

  • \(\lambda(1)=1\)

  • \(\lambda(2)=-1\)

  • \(\lambda(4)=\lambda(2^2)=1\)

  • \(\lambda(12)=\lambda(2^2\cdot 3)=-1\)

Key properties

  • Completely multiplicative: \(\lambda(ab)=\lambda(a)\lambda(b)\) for all \(a,b\).

  • Connected to the Möbius function, but without the “squarefree = 0” behavior.

Summatory function experiments

A standard experiment is the summatory function

\[ L(x)=\sum_{n\le x}\lambda(n), \]

which oscillates and has deep connections to primes and zeta-function methods.

Experiment ideas

  • compare \(L(x)\) with a random walk

  • compare \(\lambda(n)\) and \(\mu(n)\) on squarefree numbers

  • visualize \(\lambda(n)\) as a \(\pm1\) texture over the integer grid