Completely multiplicative function - Revision history

Vipul at 19:36, 6 May 2009

2009-05-06T19:36:51Z

← Older revision		Revision as of 19:36, 6 May 2009
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	==Definition==		==Definition==

	Suppose <math>f</math> is an [[arithmetic function]]. In other words, <math>f</math> is a function from the natural numbers to some [[commutative unital ring]] <math>R</math>. Then, <math>f</math> is termed '''completely multiplicative''' if it is a monoid homomorphism from the multiplicative monoid of natural numbers to the multiplicative monoid of <math>R</math>. In other words, <math>f</math> satisfies the following two conditions:		Suppose <math>f</math> is an [[arithmetic function]]. In other words, <math>f</math> is a function from the natural numbers to some [[commutative unital ring]] <math>R</math>. Then, <math>f</math> is termed '''completely multiplicative''' or '''totally multiplicative''' if it is a monoid homomorphism from the multiplicative monoid of natural numbers to the multiplicative monoid of <math>R</math>. In other words, <math>f</math> satisfies the following two conditions:

	* <math>f(1) = 1</math>.		* <math>f(1) = 1</math>.

Vipul: /* Dirichlet series */

2009-05-06T18:43:24Z

Dirichlet series

← Older revision		Revision as of 18:43, 6 May 2009
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	===Dirichlet series===		===Dirichlet series===

			{{further\|[[Product formula for Dirichlet series of completely multiplicative function]]}}

	The Dirichlet series of a completely multiplicative function can be expressed particularly nicely in terms of the values at primes. Formally, if <math>f</math> is completely multiplicative, then:		The Dirichlet series of a completely multiplicative function can be expressed particularly nicely in terms of the values at primes. Formally, if <math>f</math> is completely multiplicative, then:

Vipul at 19:31, 2 May 2009

2009-05-02T19:31:51Z

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 * <math>f(1) = 1</math>.
 * <math>f(mn) = f(m)f(n)</math> for all natural numbers <math>m,n \in \mathbb{N}</math>. Here, the multiplication on the left happens in <math>\mathbb{N}</math> while the multiplication on the right happens in <math>R</math>.
 ==Relation with other properties==
@@ Line 14: / Line 28: @@
 * [[Stronger than::Multiplicative function]]
 * [[Stronger than::Divisibility-preserving function]]

Vipul: Created page with '{{arithmetic function property}} ==Definition== Suppose $f$ is an arithmetic function. In other words, $f$ is a function from the natural numbers to s...'

2009-04-29T01:47:47Z

Created page with '{{arithmetic function property}} ==Definition== Suppose <math>f</math> is an arithmetic function. In other words, <math>f</math> is a function from the natural numbers to s...'

New page

{{arithmetic function property}}

==Definition==

Suppose <math>f</math> is an [[arithmetic function]]. In other words, <math>f</math> is a function from the natural numbers to some [[commutative unital ring]] <math>R</math>. Then, <math>f</math> is termed '''completely multiplicative''' if it is a monoid homomorphism from the multiplicative monoid of natural numbers to the multiplicative monoid of <math>R</math>. In other words, <math>f</math> satisfies the following two conditions:

* <math>f(1) = 1</math>.
* <math>f(mn) = f(m)f(n)</math> for all natural numbers <math>m,n \in \mathbb{N}</math>. Here, the multiplication on the left happens in <math>\mathbb{N}</math> while the multiplication on the right happens in <math>R</math>.

==Relation with other properties==

===Weaker properties===

* [[Stronger than::Multiplicative function]]
* [[Stronger than::Divisibility-preserving function]]

Completely multiplicative function - Revision history

Vipul at 19:36, 6 May 2009

Vipul: /* Dirichlet series */

Vipul at 19:31, 2 May 2009

Vipul: Created page with '{{arithmetic function property}} ==Definition== Suppose f is an arithmetic function. In other words, f is a function from the natural numbers to s...'

Vipul: Created page with '{{arithmetic function property}} ==Definition== Suppose $f$ is an arithmetic function. In other words, $f$ is a function from the natural numbers to s...'