Completely multiplicative function: Difference between revisions

Revision as of 21:34, 24 September 2012

Revision as of 03:09, 3 March 2013

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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 [[MobiusMöbius function]].<ref>Apostol, p. 36</ref>

Completely multiplicative functions also satisfy a pseudo-associative law. If ''f'' is completely multiplicative then