Dirichlet product - Revision history

Vipul: /* Facts */

2009-04-22T22:06:43Z

Facts

← Older revision		Revision as of 22:06, 22 April 2009
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	Most of the functions we deal with are integer-valued. Note that there is a natural map from the integers to any commutative unital ring, and thus, any integer-valued function can be viewed as a function to <math>R</math> for any commutative unital ring. This makes most sense when the ring has characteristic zero, so that the map from integers to it is injective.		Most of the functions we deal with are integer-valued. Note that there is a natural map from the integers to any commutative unital ring, and thus, any integer-valued function can be viewed as a function to <math>R</math> for any commutative unital ring. This makes most sense when the ring has characteristic zero, so that the map from integers to it is injective.

	===~~Monoid~~ structure===		===Abelian monoid structure===

	The set of all functions from <math>\mathbb{N}</math> to <math>R</math> forms a [[monoid]] with respect to the Dirichlet product:		The set of all functions from <math>\mathbb{N}</math> to <math>R</math> forms a [[monoid]] with respect to the Dirichlet product:
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	<math>\sum_{d_1d_2d_3 = n} f(d_1)g(d_2)h(d_3)</math>.		<math>\sum_{d_1d_2d_3 = n} f(d_1)g(d_2)h(d_3)</math>.
			* [[Dirichlet product is commutative]]: The fact that <math>f * g = g * f</math> is direct from the definition, and is based on the observation that the role of the divisors <math>d</math> and <math>n/d</math> can be switched.
	* [[Identity element for Dirichlet product is indicator function for one]]: The identity element for the Dirichlet product is the function <math>I</math>, defined as <math>I(1) = 1</math>, and <math>I(n) = 0</math> for <math>n \ne 1</math>.		* [[Identity element for Dirichlet product is indicator function for one]]: The identity element for the Dirichlet product is the function <math>I</math>, defined as <math>I(1) = 1</math>, and <math>I(n) = 0</math> for <math>n \ne 1</math>.

	===~~Group~~ structure===		===Abelian group structure===

	If <math>f(1) = 1</math> (where the <math>1</math> on the left is the natural number, and the <math>1</math> on the right is the identity element of the ring), then <math>f</math> has a multiplicative inverse with respect to the Dirichlet product. Moreover, this inverse ''also'' sends <math>1</math> to <math>1</math>. The functions that send <math>1</math> to <math>1</math>, in fact, form a group under the Dirichlet product. (More generally, we can look at all functions that send <math>1</math> to a unit).		If <math>f(1) = 1</math> (where the <math>1</math> on the left is the natural number, and the <math>1</math> on the right is the identity element of the ring), then <math>f</math> has a multiplicative inverse with respect to the Dirichlet product. Moreover, this inverse ''also'' sends <math>1</math> to <math>1</math>. The functions that send <math>1</math> to <math>1</math>, in fact, form a group under the Dirichlet product. (More generally, we can look at all functions that send <math>1</math> to a unit).

Vipul: /* Facts */

2009-04-22T19:42:20Z

Facts

← Older revision		Revision as of 19:42, 22 April 2009
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	An important subgroup of this group is the group of all [[multiplicative function]]s. A Dirichlet product of multiplicative functions is multiplicative, and the inverse of a multiplicative function is multiplicative. {{further\|[[Multiplicative functions form a group under Dirichlet product]]}}		An important subgroup of this group is the group of all [[multiplicative function]]s. A Dirichlet product of multiplicative functions is multiplicative, and the inverse of a multiplicative function is multiplicative. {{further\|[[Multiplicative functions form a group under Dirichlet product]]}}

			===Important functions===

			A complete list of commonly studied arithmetic functions is at:

			[[:Category:Arithmetic functions]]

			Some particular important ones are:

			* The identity element for Dirichlet product: Denoted <math>I</math>, this is the ''indicator function'' for <math>1</math>: it is <math>1</math> at <math>1</math> and <math>0</math> elsewhere.
			* The [[all ones function]]: This function sends everything to <math>1</math>. This is denoted by <math>U</math>. Note that although this is the identity for ''pointwise'' multiplication, it is ''not'' the identity for the Dirichlet product.
			* The [[Mobius function]]: Denoted <math>\mu</math>, this is the inverse of the all ones function with respect to the Dirichlet product.
			* The [[identity function]]: This function sends every natural number to itself, now viewed as a ring element. This is denoted <math>E</math>. Although this would be the identity for ''composition'', it is ''not'' the identity for the Dirichlet product.

Vipul: Created page with '==Definition== Suppose $\mathbb{N}$ is the set of natural numbers and $R$ is a commutative unital ring. Suppose $f,g: \mathbb{N} \to R$ ...'

2009-04-22T19:37:09Z

Created page with '==Definition== Suppose <math>\mathbb{N}</math> is the set of natural numbers and <math>R</math> is a commutative unital ring. Suppose <math>f,g: \mathbb{N} \to R</math> ...'

New page

==Definition==

Suppose <math>\mathbb{N}</math> is the set of [[natural number]]s and <math>R</math> is a [[commutative unital ring]]. Suppose <math>f,g: \mathbb{N} \to R</math> are two functions. The '''Dirichlet product''' or '''Dirichlet convolution''' of <math>f</math> and <math>g</math>, denoted <math>(f * g)</math>, is defined as:

<math>(f * g)(n) = \sum_{d|n} f(d)g(n/d)</math>.

The sum is over all ''positive integers'' dividing <math>n</math>. Equivalently, it can be written as:

<math>(f * g)(n) = \sum_{d_1d_2 = n} f(d_1)g(d_2)</math>.

Here, the summation is restricted to the cases where both <math>d_1</math> and <math>d_2</math> are positive integers.

==Facts==

===Ignoring the ring===

Most of the functions we deal with are integer-valued. Note that there is a natural map from the integers to any commutative unital ring, and thus, any integer-valued function can be viewed as a function to <math>R</math> for any commutative unital ring. This makes most sense when the ring has characteristic zero, so that the map from integers to it is injective.

===Monoid structure===

The set of all functions from <math>\mathbb{N}</math> to <math>R</math> forms a [[monoid]] with respect to the Dirichlet product:

* [[Dirichlet product is associative]]: We can see that <math>f * (g * h) = (f * g) * h</math>, and both are equal to:

<math>\sum_{d_1d_2d_3 = n} f(d_1)g(d_2)h(d_3)</math>.

* [[Identity element for Dirichlet product is indicator function for one]]: The identity element for the Dirichlet product is the function <math>I</math>, defined as <math>I(1) = 1</math>, and <math>I(n) = 0</math> for <math>n \ne 1</math>.

===Group structure===

If <math>f(1) = 1</math> (where the <math>1</math> on the left is the natural number, and the <math>1</math> on the right is the identity element of the ring), then <math>f</math> has a multiplicative inverse with respect to the Dirichlet product. Moreover, this inverse ''also'' sends <math>1</math> to <math>1</math>. The functions that send <math>1</math> to <math>1</math>, in fact, form a group under the Dirichlet product. (More generally, we can look at all functions that send <math>1</math> to a unit).

An important subgroup of this group is the group of all [[multiplicative function]]s. A Dirichlet product of multiplicative functions is multiplicative, and the inverse of a multiplicative function is multiplicative. {{further|[[Multiplicative functions form a group under Dirichlet product]]}}

Dirichlet product - Revision history

Vipul: /* Facts */

Vipul: /* Facts */

Vipul: Created page with '==Definition== Suppose \mathbb{N} is the set of natural numbers and R is a commutative unital ring. Suppose f,g: \mathbb{N} \to R ...'

Vipul: Created page with '==Definition== Suppose $\mathbb{N}$ is the set of natural numbers and $R$ is a commutative unital ring. Suppose $f,g: \mathbb{N} \to R$ ...'