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→Properties: The "pseudo-associative law" which is referenced is a very well-known algebraic property in mathematics called the "distributive law". It is a property of rings that the multiplication operator distributes over the addition operator. |
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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>
Completely multiplicative functions also satisfy a
<math>f \cdot (g*h)=(f \cdot g)*(f \cdot h)</math>
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Here <math> \tau</math> is the [[divisor function]].
===Proof of
:<math>
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