# 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>, 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>. | |

− | + | 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. | |

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+ | 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( | ||

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## Revision as of 21:55, 6 April 2008

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; the Mobius inversion formula relates these two quantities.

Examples in elementary number theory include the identity map, the number of divisors, the sum of divisors (and its generalization , the Euler phi function, the number of divisors (also denoted , $\mu(
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