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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. Arithmetic functions which can be written as the Dirichlet convolution of two completely multiplicative functions are said to be quadratics or specially multiplicative multiplicative functions. They are rational arithmetic functions of order (2, 0) and obey the Busche-Ramanujan identity.
There are a variety of statements about a function which are equivalent to it being completely multiplicative. For example, if a function ''f'' is multiplicative then it is completely multiplicative if and only if its [[Dirichlet inverse]] is <math>\mu\cdot f</math> where <math>\mu</math> is the [[Möbius function]].<ref>Apostol, p. 36</ref>
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==References==
<references />
* T. M. Apostol, Some properties of completely multiplicative arithmetical functions, Amer. Math. Monthly 78 (1971) 266-271.
* P. Haukkanen, On characterizations of completely multiplicative arithmetical functions, in Number theory, Turku, de Gruyter, 2001, pp. 115–123.
* E. Langford, Distributivity over the Dirichlet product and completely multiplicative arithmetical functions, Amer. Math. Monthly 80 (1973) 411–414.
* V. Laohakosol, Logarithmic operators and characterizations of completely multiplicative functions, Southeast Asian Bull. Math. 25 (2001) no. 2, 273–281.
* K. L. Yocom, Totally multiplicative functions in regular convolution rings, Canad. Math. Bull. 16 (1973) 119–128.
[[Category:Multiplicative functions]]
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