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

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Multiplicative arithmetic functions

Definition

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

$f (1) = 1$

and

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

for every pair of coprime positive integers $m, n$ (Coprime integers: $\gcd (a, b) = 1$ ).

Remarks

The normalization $f (1) = 1$ is part of the definition. It excludes the identically zero convention used in some other contexts.

Depends on

Used by

Dependency tree · two levels

9 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

Victor Shoup, A Computational Introduction to Number Theory and Algebra, Section 2.9 (standard reference, not scraped)
Kiran S. Kedlaya, An Introduction to Analytic Number Theory, Definition 3.6 (standard reference, not scraped)