Convolution
For two functions , the Dirichlet convolution (or simply convolution)
of
and
is defined as
.
We usually only consider positive divisors of . We are often interested in convolutions of weak multiplicative functions; the set of weak multiplicative functions is closed under convolution. In general, convolution is commutative and associative; it also has an identity, the function
defined to be 1 if
, and 0 otherwise. However, not all functions have inverses (e.g., the function
has no inverse, as
, for all functions
), although many do.
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