Arithmetic function

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Revision as of 06:03, 15 June 2009 by 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 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