Difference between revisions of "Multiplicative function"
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− | A '''multiplicative function''' <math>f : S \to T</math> is a [[function]] which [[commute]]s with multiplication. That is, <math>S</math> and <math>T</math> must be [[set]]s with multiplication such that <math>f(x\cdot y) = f(x) \cdot f(y)</math> for all <math>x, y \in S</math>. | + | A '''multiplicative function''' <math>f : S \to T</math> is a [[function]] which [[commute]]s with multiplication. That is, <math>S</math> and <math>T</math> must be [[set]]s with multiplication such that <math>f(x\cdot y) = f(x) \cdot f(y)</math> for all <math>x, y \in S</math>, i.e. it preserves the multiplicative structure. A prominent special case of this would be a homomorphism between groups, which preserves the whole group structure (inverses and identity in addition to multiplication). |
Most frequently, one deals with multiplicative functions <math>f : \mathbb{Z}_{>0} \to \mathbb{C}</math>. These functions appear frequently in [[number theory]], especially in [[analytic number theory]]. In this case, one sometimes also defines ''weak multiplicative functions'': a function <math>f: \mathbb{Z}_{>0} \to \mathbb{C}</math> is weak multiplicative if and only if <math>f(mn) = f(m)f(n)</math> for all pairs of [[relatively prime]] [[integer]]s <math>(m, n)</math>. | Most frequently, one deals with multiplicative functions <math>f : \mathbb{Z}_{>0} \to \mathbb{C}</math>. These functions appear frequently in [[number theory]], especially in [[analytic number theory]]. In this case, one sometimes also defines ''weak multiplicative functions'': a function <math>f: \mathbb{Z}_{>0} \to \mathbb{C}</math> is weak multiplicative if and only if <math>f(mn) = f(m)f(n)</math> for all pairs of [[relatively prime]] [[integer]]s <math>(m, n)</math>. |
Revision as of 00:02, 3 April 2008
This is an AoPSWiki Word of the Week for March 28-April 5 |
A multiplicative function is a function which commutes with multiplication. That is, and must be sets with multiplication such that for all , i.e. it preserves the multiplicative structure. A prominent special case of this would be a homomorphism between groups, which preserves the whole group structure (inverses and identity in addition to multiplication).
Most frequently, one deals with multiplicative functions . These functions appear frequently in number theory, especially in analytic number theory. In this case, one sometimes also defines weak multiplicative functions: a function is weak multiplicative if and only if for all pairs of relatively prime integers .
Let and be multiplicative in the number theoretic sense ("weak multiplicative"). Then the function of defined by is also multiplicative. This article is a stub. Help us out by expanding it.