Difference between revisions of "Multiplicative function"
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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. | 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, <math>\phi(n)</math> the Euler phi function, <math>\tau(n)</math> ( | + | 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 17:59, 4 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 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( This article is a stub. Help us out by expanding it.