Additive function: Difference between revisions

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Different definitions exist depending on the specific field of application. Traditionally, an '''additive function''' 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.

In [[number theory]], an '''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:

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

Outside number theory, the term '''additive''' ismay usuallyalso be used for all functions with the property ''f''(''ab'') = ''f''(''a'') + ''f''(''b'') for all arguments ''a'' and ''b''. This article discusses number theoretic additive functions.

An additive function may also refer to an [[additive polynomial]]. In this case, ''f'' is taken to be a function on a [[field (mathematics)|field]], and additivity is taken to be the property that

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

for any two elements ''x'' and ''y'' of the field. This article covers only the first definition; see the article [[additive polynomial]] for the second. Note also that any [[homomorphism]] ''f'' between [[Abelian group]]s is "additive" by the second definition.

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

== Completely additive ==

forFor anya twospecific elements ''x'' and ''y''case of the field. This article covers only the first definition; see the article [[additive polynomial]] for the second. Note also that any [[homomorphism]] ''f'' between [[Abelian group]]s is "additive" by the secondfirst definition.

== Completely additive ==

Outside number theory, the term '''additive''' is usually used for all functions with the property ''f''(''ab'') = ''f''(''a'') + ''f''(''b'') for all arguments ''a'' and ''b''. This article discusses number theoretic additive functions.

An additive function ''f''(''n'') is said to be '''completely additive''' if ''f''(''ab'') = ''f''(''a'') + ''f''(''b'') holds ''for all'' positive integers ''a'' and ''b'', even when they are not coprime.