A note on notation:
are arithmetic functions, then
is the Dirichlet product thereof, such that
. Theorem: If both and are multiplicative, then is multiplicative. Proof:
We shall prove by contradiction. Suppose
is not multiplicative. Let
is not multiplicative, there exist
. We choose
as small as possible. If
is not multiplicative, a contradiction. If
, we have
is not multiplicative, a contradiction.
Okay, here's my problem with this "proof": Why in the world do we have " for all and " if we are assuming that is not multiplicative? As far as I can see, such a statement might be true or false for some . Yet the author of the proof takes it to be true. Why?