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Added definition in logic notation & further explanation for f(1) = 1 condition. |
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A completely multiplicative function (or totally multiplicative function) is an [[arithmetic function]] (that is, a function whose [[Domain of a function|___domain]] is the [[natural number]]s), such that ''f''(1) = 1 and ''f''(''ab'') = ''f''(''a'')''f''(''b'') holds ''for all'' positive integers ''a'' and ''b''.<ref>{{cite book|last=Apostol|first=Tom|title=Introduction to Analytic Number Theory|year=1976|publisher=Springer|isbn=0-387-90163-9|pages=[https://archive.org/details/introductiontoan00apos_0/page/30 30]|url-access=registration|url=https://archive.org/details/introductiontoan00apos_0/page/30}}</ref>
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>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</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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