Difference between revisions of "Multiplicative function"
(wotw) |
|||
Line 4: | Line 4: | ||
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>. | ||
+ | |||
+ | Let <math>f(n)</math> and <math>g(n)</math> be multiplicative in the number theoretic sense ("weak multiplicative"). Then the function of <math>n</math> defined by <cmath>\sum_{d|n} f(d) g(\frac{n}{d})</cmath> is also multiplicative. | ||
{{stub}} | {{stub}} |
Revision as of 17:44, 31 March 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 .
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.