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