Multiplicative function: Difference between revisions

Revision as of 01:53, 29 April 2009

Let $f$ be an arithmetic function: in other words, $f$ is a function from the set of natural numbers to a commutative unital ring $R$ . We say that $f$ is multiplicative if it satisfies the following two conditions:

$f(1)=1$ .
$f(mn)=f(m)f(n)$ for all pairs of relatively prime numbers $m,n\in \mathbb {N}$ .

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 ==Definition==
-Let <math>f</math> be a function from the set of [[natural number]]s to a [[commutative unital ring]] <math>R</math>. We say that <math>f</math> is '''multiplicative''' if, whenever <math>m</math> and <math>n</math> are relatively prime natural numbers, we have:
+Let <math>f</math> be an [[arithmetic function]]: in other words, <math>f</math> is a function from the set of [[natural number]]s to a [[commutative unital ring]] <math>R</math>. We say that <math>f</math> is '''multiplicative''' if it satisfies the following two conditions:
-<math>f(mn) = f(m)f(n)</math>.
+* <math>f(1) = 1</math>.
+* <math>f(mn) = f(m)f(n)</math> for all pairs of relatively prime numbers <math>m,n \in \mathbb{N}</math>.
 ==Relation with other properties==
@@ Line 9: / Line 10: @@
 ===Stronger properties===
-* [[Weaker than::completely multiplicative function]]
+* [[Weaker than::Completely multiplicative function]]
+===Incomparable properties===
+* [[Divisibility-preserving function]]