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 23:02, 2 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 ![\[\sum_{d|n} f(d) g(\frac{n}{d})\]](http://latex.artofproblemsolving.com/f/6/d/f6d32731d20bf69708c84b2c78dab495742a2498.png)
is also multiplicative.
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