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→Examples: the set of square-free numbers is not closed under multiplication (for exemple 6×10=60=4×15) as was sugested by the original text. Although he function constructed from it really is multiplicative, as was suggested before. Tags: Mobile edit Mobile app edit Android app edit App section source |
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** Id<sub>1</sub>(''n'') = Id(''n'').
* ''ε''(''n''): the function defined by ''ε''(''n'') = 1 if ''n'' = 1 and 0 otherwise, sometimes called ''multiplication unit for [[Dirichlet convolution]]'' or simply the ''[[unit function]]'' (completely multiplicative). Sometimes written as ''u''(''n''), but not to be confused with ''μ''(''n'') .
* 1<sub>''C''</sub>(''n''), the [[indicator function]] of the set ''C'' ⊂ '''Z''', for certain sets ''C''. The indicator function 1<sub>''C''</sub>(''n'') is multiplicative precisely when the set ''C'' has the following property for any coprime numbers ''a'' and ''b'': the product ''ab'' is in ''C'' if and only if the numbers ''a'' and ''b'' are both themselves in ''C''. This is the case if ''C'' is the set of squares, cubes, or ''k''-th powers
Other examples of multiplicative functions include many functions of importance in number theory, such as:
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