Arithmetic function: Difference between revisions
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In [[number theory]], an '''arithmetic function''' is a function defined on the set of [[ | In [[number theory]], an '''arithmetic function''' is a function defined on the set of [[positive integer]]s, usually with [[integer]], [[real number|real]] or [[complex number|complex]] values. | ||
==Classes of arithmetic function== | ==Classes of arithmetic function== | ||
Arithmetic functions which have some connexion with the additive or multiplicative structure of the integers are of particular interest in number theory. | Arithmetic functions which have some connexion with the additive or multiplicative structure of the integers are of particular interest in number theory. | ||
===Multiplicative functions=== | |||
We define a function ''a''(''n'') on positive integers to be | |||
* '''Totally multiplicative''' if <math>a(mn) = a(m) a(n)</math> for all ''m'' and ''n''. | * '''Totally multiplicative''' if <math>a(mn) = a(m) a(n)</math> for all ''m'' and ''n''. | ||
* '''Multiplicative''' if <math>a(mn) = a(m) a(n)</math> whenever ''m'' and ''n'' are [[coprime]]. | * '''Multiplicative''' if <math>a(mn) = a(m) a(n)</math> whenever ''m'' and ''n'' are [[coprime]]. | ||
The ''[[Dirichlet convolution]]'' of two arithmetic function ''a''(''n'') and ''b''(''n'') is defined as | |||
:<math>a \star b (n) = \sum_{d \mid n} a(d) b(n/d) .\,</math> | |||
If ''a'' and ''b'' are multiplicative, so is their convolution. | |||
==Examples== | ==Examples== | ||
* Carmichael's [[lambda function]] | * Carmichael's [[lambda function]] | ||
* A [[Dirichlet character]] | |||
* [[Euler]]'s [[totient function]] | * [[Euler]]'s [[totient function]] | ||
* [[Jordan's totient function]] | * [[Jordan's totient function]] | ||
* [[Möbius function]] | * [[Möbius function]][[Category:Suggestion Bot Tag]] | ||
Latest revision as of 16:00, 12 July 2024
In number theory, an arithmetic function is a function defined on the set of positive integers, usually with integer, real or complex values.
Classes of arithmetic function
Arithmetic functions which have some connexion with the additive or multiplicative structure of the integers are of particular interest in number theory.
Multiplicative functions
We define a function a(n) on positive integers to be
- Totally multiplicative if for all m and n.
- Multiplicative if whenever m and n are coprime.
The Dirichlet convolution of two arithmetic function a(n) and b(n) is defined as
If a and b are multiplicative, so is their convolution.