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==Properties==
A completely multiplicative function is completely determined by its values at the prime numbers, a consequence of the [[fundamental theorem of arithmetic]]. Thus, if ''n'' is a product of powers of distinct primes, say ''n'' = ''p''<sup>''a''</sup> ''q''<sup>''b''</sup> ..., then ''f''(''n'') = ''f''(''p'')<sup>''a''</sup> ''f''(''q'')<sup>''b''</sup> ...
While the [[Dirichlet convolution]] of two multiplicative functions is multiplicative, the [[Dirichlet convolution]] of two completely multiplicative functions need not be completely multiplicative.
There are a variety of statements about a function which are equivalent to it being completely multiplicative. For example, if a function ''f'' multiplicative then is completely multiplicative if and only if the [[Dirichlet inverse]] is <math>\mu f</math> where <math>\mu</math> is the [[Mobius function]].<ref>Apostol, p. 36</ref>
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