Multiplicative function
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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