Multiplicative function: Difference between revisions

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{{hatnote|Outside number theory, the term '''multiplicative function''' is usually used for [[completely multiplicative function]]s. This article discusses number theoretic multiplicative functions.}}
 
In [[number theory]], a '''multiplicative function''' is an [[arithmetic function]] ''<math>f''(''n'')</math> of a positive [[integer]] ''<math>n''</math> with the property that ''<math>f''(1) = 1</math> and
<math display="block">f(ab) = f(a)f(b)</math> whenever ''<math>a''</math> and ''<math>b''</math> are [[coprime]].
 
An arithmetic function ''f''(''n'') is said to be '''[[completely multiplicative function|completely multiplicative]]''' (or '''totally multiplicative''') if ''<math>f''(1) = 1</math> and ''<math>f''(''ab'') = ''f''(''a'')''f''(''b'')</math> holds ''for all'' positive integers ''<math>a''</math> and ''<math>b''</math>, even when they are not coprime.
 
== Examples ==