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[edit]

Arithmetic functions which have some connexion with the additive or multiplicative structure of the integers are of particular interest in number theory.

Multiplicative functions[edit]

We define a function a(n) on positive integers to be

• Totally multiplicative if ${\displaystyle a(mn)=a(m)a(n)}$ for all m and n.
• Multiplicative if ${\displaystyle a(mn)=a(m)a(n)}$ whenever m and n are coprime.

The Dirichlet convolution of two arithmetic function a(n) and b(n) is defined as

${\displaystyle a\star b(n)=\sum _{d\mid n}a(d)b(n/d).\,}$

If a and b are multiplicative, so is their convolution.

Examples[edit]

• Carmichael's lambda function
• A Dirichlet character
• Euler's totient function
• Jordan's totient function
• Möbius function