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

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Completely multiplicative arithmetic functions

Definition

An arithmetic function $f$ (Arithmetic functions on the positive integers) is completely multiplicative when

$f (1) = 1$

and

$f (m n) = f (m) f (n)$

for all positive integers $m, n$ .

Remarks

Every completely multiplicative arithmetic function is multiplicative, because the displayed rule already applies when $m$ and $n$ are coprime.

Depends on

Arithmetic functions on the positive integers

Used by

Tau is multiplicative but not completely multiplicative Counterexample
The divisor functions arise by Dirichlet convolution Proposition

Dependency tree · two levels

3 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

Kiran S. Kedlaya, An Introduction to Analytic Number Theory, Definition 3.7 (standard reference, not scraped)
Karl-Dieter Crisman, Number Theory: In Context and Interactive, Section 23.3 (standard reference, not scraped)