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Added anchors to algebraic and number theoretic definitions of "additive function" |
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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]].
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'').
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