Arithmetic function: Difference between revisions

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imported>Richard Pinch
imported>Jitse Niesen
(remove "See also" section because it is subsumed in the "Related Articles" subpage; perhaps the "Examples" section should go for the same reason, though it may be better to flesh it out a bit)
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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.   
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==
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* [[Jordan's totient function]]
* [[Jordan's totient function]]
* [[Möbius function]]
* [[Möbius function]]
==See also==
* [[Formal Dirichlet series]]
* [[Average order of an arithmetic function]]
* [[Normal order of an arithmetic function]]

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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.

Examples