Difference between revisions of "Multiplicative function"
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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. | + | 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, <math>\phi(n)</math> the Euler phi function, <math>\tau(n)</math> (I actually forget what this is), <math>\mu(n)</math> the Mobius function, and bunches of other totally awesome stuff. | ||
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Revision as of 00:04, 3 April 2008
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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; 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, 
the Euler phi function, 
(I actually forget what this is), 
the Mobius function, and bunches of other totally awesome stuff.
This article is a stub. Help us out by expanding it.
