Multiplicative function
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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,
the Euler phi function,
(I actually forget what this is),
the Mobius function, and bunches of other totally awesome stuff.
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