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Adding local short description: "Arithmetic function", overriding Wikidata description "arithmetic function that is multiplicative for every pair of integers, not necessarily coprime" |
Dedhert.Jr (talk | contribs) →Definition: cleanup |
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In logic notation: <math>f(1) = 1</math> and <math>\forall a, b \in \text{___domain}(f), f(ab) = f(a)(b)</math>.
Without the requirement that ''f''(1) = 1, one could still have ''f''(1) = 0, but then ''f''(''a'') = 0 for all positive integers ''a'', so this is not a very strong restriction. If one did not fix <math>f(1) = 1</math>, one can see that both <math>0</math> and <math>1</math> are possibilities for the value of <math>f(1)</math> in the following way:
<math display="block"> \begin{align} f(1) = f(1 \cdot 1) &\iff f(1) = f(1)f(1) \\ &\iff f(1) = f(1)^2 \\ &\iff f(1)^2 - f(1) = 0 \\ &\iff f(1)\left(f(1) - 1\right) = 0 \\ &\iff f(1) = 0 \lor f(1) = 1 \end{align} </math>
The definition above can be rephrased using the language of algebra: A completely multiplicative function is a [[homomorphism]] from the [[monoid]] <math>(\mathbb Z^+,\cdot)</math> (that is, the positive integers under multiplication) to some other monoid.
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