Additive function: Difference between revisions

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{{more footnotes|date=February 2013}}

{{About||the [[Abstract algebra|algebra]]ic meaning|Additive map}}

In [[mathematics]] the term '''additive function''' has two different definitions, depending on the specific field of application.

In [[algebra]] an '''{{anchor|definition-additive_function-algebra}}additive function''' (or '''additive map''') is a function that preserves the addition operation:

:''f''(''x'' + ''y'') = ''f''(''x'') + ''f''(''y'')

for any two elements ''x'' and ''y'' in the ___domain. For example, any [[linear map]] is additive. When the ___domain is the [[real numbers]], this is [[Cauchy's functional equation]]. For a specific case of this definition, see [[additive polynomial]]. Any [[homomorphism]] ''f'' between [[abelian group]]s is additive by this definition.

In [[number theory]], an '''{{anchor|definition-additive_function-number_theory}}additive function''' is an [[arithmetic function]] ''f''(''n'') of the positive [[integer]] ''n'' such that whenever ''a'' and ''b'' are [[coprime]], the function of the product is the sum of the functions:<ref name="Erdos1939">Erdös, P., and M. Kac. On the Gaussian Law of Errors in the Theory of Additive Functions. Proc Natl Acad Sci USA. 1939 April; 25(4): 206–207. [http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1077746/ online]</ref>

:''f''(''ab'') = ''f''(''a'') + ''f''(''b'').

The remainder of this article discusses number theoretic additive functions, using the second definition.

== Completely additive ==