Multiplicative function - Revision history

Vipul: /* Effect of operations */

2009-05-06T19:35:10Z

Effect of operations

← Older revision		Revision as of 19:35, 6 May 2009
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	===Dirichlet product===		===Dirichlet product===

	{{further\|[[Multiplicative functions form a group under Dirichlet product]]}}		{{further\|[[Multiplicative functions form abelian group under Dirichlet product]]}}

	Under the Dirichlet product, the multiplicative functions form a group. In other words, the Dirichlet product of two multiplicative functions is multiplicative, and the inverse of a multiplicative function with respect to the Dirichlet product is also a multiplicative function.		Under the [[Dirichlet product]], the multiplicative functions form an [[abelian group]]. In other words, the Dirichlet product of two multiplicative functions is multiplicative, and the inverse of a multiplicative function with respect to the Dirichlet product is also a multiplicative function.

	===Pointwise product===		===Pointwise product===

Vipul at 19:41, 2 May 2009

2009-05-02T19:41:10Z

← Older revision		Revision as of 19:41, 2 May 2009
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	* [[Divisibility-preserving function]]		* [[Divisibility-preserving function]]

			==Effect of operations==

			===Dirichlet product===

			{{further\|[[Multiplicative functions form a group under Dirichlet product]]}}

			Under the Dirichlet product, the multiplicative functions form a group. In other words, the Dirichlet product of two multiplicative functions is multiplicative, and the inverse of a multiplicative function with respect to the Dirichlet product is also a multiplicative function.

			===Pointwise product===

			The pointwise product of two multiplicative functions is a multiplicative function. Note that a multiplicative function is invertible with respect to the pointwise product if and only if its value at every natural number is invertible, in which case its inverse is also a multiplicative function. In particular, a multiplicative function that is zero at any natural number cannot be invertible with respect to the pointwise product.

			===Composition===

			If <math>f,g</math> are multiplicative functions such that <math>g</math> takes only positive integer values, it is ''not'' necessary that <math>f \circ g</math> be a multiplicative function. The reason is that <math>g(m)</math> and <math>g(n)</math> need not be relatively prime even if <math>m</math> and <math>n</math> are relatively prime.

			However, it is true that if <math>f</math> is completely multiplicative and <math>g</math> is multiplicative with positive integer values, then <math>f \circ g</math> is multiplicative. {{further\|[[Composite of completely multiplicative function and multiplicative function is multiplicative]]}}

Vipul at 19:36, 2 May 2009

2009-05-02T19:36:08Z

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 * <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==

Vipul at 01:53, 29 April 2009

2009-04-29T01:53:20Z

← Older revision		Revision as of 01:53, 29 April 2009
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	==Definition==		==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==		==Relation with other properties==
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	===Stronger properties===		===Stronger properties===

	* [[Weaker than::~~completely~~ multiplicative function]]		* [[Weaker than::Completely multiplicative function]]

			===Incomparable properties===

			* [[Divisibility-preserving function]]

Vipul: Created page with '==Definition== Let $f$ be a function from the set of natural numbers to a commutative unital ring $R$ . We say that $f$ is '''multiplicat...'

2009-04-28T23:12:10Z

Created page with '==Definition== Let <math>f</math> be a function from the set of natural numbers to a commutative unital ring <math>R</math>. We say that <math>f</math> is '''multiplicat...'

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