Difference between revisions of "Multiplicative function"

m
(wotw)
Line 1: Line 1:
{{stub}}
+
{{WotWAnnounce|week=March 28-April 5}}
  
 
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>.
  
 
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>.
 +
{{stub}}

Revision as of 20:12, 28 March 2008

This is an AoPSWiki Word of the Week for March 28-April 5

A multiplicative function $f : S \to T$ is a function which commutes with multiplication. That is, $S$ and $T$ must be sets with multiplication such that $f(x\cdot y) = f(x) \cdot f(y)$ for all $x, y \in S$.

Most frequently, one deals with multiplicative functions $f : \mathbb{Z}_{>0} \to \mathbb{C}$. These functions appear frequently in number theory, especially in analytic number theory. In this case, one sometimes also defines weak multiplicative functions: a function $f: \mathbb{Z}_{>0} \to \mathbb{C}$ is weak multiplicative if and only if $f(mn) = f(m)f(n)$ for all pairs of relatively prime integers $(m, n)$. This article is a stub. Help us out by expanding it.

Invalid username
Login to AoPS