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In [[mathematics]], especially in the field of [[number theory]], functions from the natural numbers which respect products are important, and are given the name '''completely multiplicative functions'''. In number theory, a weaker condition is also important, respecting only products of [[coprime]] numbers, and such functions are called [[multiplicative function]]s. Outside of number theory, the term "multiplicative function" is often taken to be synonymous with "completely multiplicative function" as defined in this article.
#REDIRECT [[Multiplicative function]]▼
==Definition==
A '''completely multiplicative function''' is an [[arithmetic function]] (that is, a function whose [[___domain]] is the [[natural number]]s), such that ''f''(1) = 1 and ''f''(''ab'') = ''f''(''a'') ''f''(''b'') holds ''for all'' positive integers ''a'' and ''b''.
Without the requirement that ''f''(1) = 1, one could still have ''f''(1) = 0, but then ''f''(''a'') = 0 for all positive integers ''a'', so this is not a very strong restriction.
==Examples==
The easiest example of a multiplicative function is a [[monomial]]: For any particular positive integer ''n'', define ''f''(''a'') = ''a''<sup>''n''</sup>.
==See also==
[[Category:Multiplicative functions]]
{{numtheory-stub}}
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