DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-31

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Liouville's function

Definition

For $n \geq 1$ , list its distinct prime divisors as $p_{0}, \dots, p_{r - 1}$ and put $e_{i} : = v_{p_{i}} (n) \geq 1$ . Its canonical prime factorization is

n = \prod_{i < r} p_{i}^{e_{i}} .

The Liouville function is

λ (n) : = (- 1)^{e_{0} + \dots + e_{r - 1}} .

For $n = 1$ the exponent sum is empty, so $λ (1) = 1$ .

Remarks

For $n \geq 1$ and any injective list $p : r \to Z$ of primes containing every prime divisor of $n$ , one has $n = \prod_{i < r} p_{i}^{v_{p_{i}} (n)}$ ; the exponents are determined by $n$ , and $v_{q} (n) = 0$ for every prime $q$ outside the list makes the positive exponent attached to each prime divisor unique. Reordering the distinct prime divisors does not change their sum, so the parity of $e_{0} + \dots + e_{r - 1}$ is well defined.

Depends on

Used by

Working out the square-indicator convolution for prime powers Example
The convolution 1*λ detects perfect squares Proposition

Dependency tree · two levels

31 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Karl-Dieter Crisman, Number Theory: In Context and Interactive, Definition 23.3.4 (standard reference, not scraped)
Tom Sanders, Topics in Analytic Number Theory, Chapter 1 (standard reference, not scraped)