For two functions , the Dirichlet convolution (or simply convolution, when the context is clear) 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. Not all functions have inverses (e.g., the function has no inverse, as , for all functions ), although all functions such that have inverses.
Closure of Weak Multiplicative Functions Under Convolution
Theorem. If are weak multiplicative functions, then so is .
Proof. Let be relatively prime. We wish to prove that .
For , let be the set of divisors of . For relatively prime , we claim that the function is a bijection from to . Indeed, for any and , so . Furthermore, for each , there exist unique such that , , . Thus is bijective. As a result of our claim, we have the identity
for any functions mapping subsets of into . In particular, we may let the domains of and be , and define and . We then have
But since each divisor of is relatively prime to every divisor of , we have
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