Difference between revisions of "Multiplicative function"

(wotw)
(rm)
(4 intermediate revisions by 3 users not shown)
Line 1: Line 1:
{{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>, 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).
  
A '''multiplicative function''' <math>f : S \to T</math> is a [[function]] which [[commute]]s with multiplicationThat 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>.
+
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; the Mobius inversion formula relates these two quantities.
 +
 
 +
Examples in elementary number theory include the identity map, <math>d(n)</math> the number of divisors, <math>\sigma(n)</math> the sum of divisors (and its generalization <math>\sigma_k(n) = \sum_{d|n}d^k</math>, <math>\phi(n)</math> the Euler phi function, <math>\tau(n)</math> the number of divisors (also denoted <math>\sigma_0(n)</math>, $\mu(
 
{{stub}}
 
{{stub}}

Revision as of 21:55, 6 April 2008

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 $x, y \in S$, 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 $f : \mathbb{Z}_{>0} \to \mathbb{C}$. 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 $(m, n)$.

Let $f(n)$ and $g(n)$ be multiplicative in the number theoretic sense ("weak multiplicative"). Then the function of $n$ defined by \[\sum_{d|n} f(d) g(\frac{n}{d})\] is also multiplicative; the Mobius inversion formula relates these two quantities.

Examples in elementary number theory include the identity map, $d(n)$ the number of divisors, $\sigma(n)$ the sum of divisors (and its generalization $\sigma_k(n) = \sum_{d|n}d^k$, $\phi(n)$ the Euler phi function, $\tau(n)$ the number of divisors (also denoted $\sigma_0(n)$, $\mu( This article is a stub. Help us out by expanding it.

Invalid username
Login to AoPS