Section 24.1 Products and Sums
In order to motivate bringing infinite processes to this part of number theory, let’s step back a bit. Many functions we have already seen may be thought of in two ways – either as a product or as a sum.
Subsection 24.1.1 Products
Let as an indexing tool denote the set of primes which divide (recall Example 6.3.4). Then we have the following product representation of two familiar arithmetic functions. (Recall Theorem 19.2.5 and Fact 18.1.1.)
Both of these functions therefore may be thought of as (finite) products.
As a related example, we explicitly wrote out the product for the abundancy index in Section 19.3.
Alternately, to avoid fractions:
Note that
Subsection 24.1.2 Products that are sums
On the other hand, these products over primes are also sums over divisors; this is true either by definition or by theorem, depending on how you look at it.
We can even cleverly add up the divisors in the opposite order to get the slightly more felicitous
With we have something to prove to make this connection, but not much. In Fact 23.3.2 we saw that Equivalently, we have Möbius-inverted Fact 9.5.4 to obtain, from the formula
By adding the divisors in the opposite order (alternately, by noting is commutative) we can write
which allows us to also write the fraction as
Now, in some sense we already knew all this. Great, some arithmetic functions can be represented either as a sum over divisors or as a product over primes, depending on what you need from them. So what?
The genius of Euler was to directly connect these ideas.
Well, this was almost the genius; his real genius was to take these ideas to the limit!
One can’t really take these expressions to infinity as they stand – one would get massive divergence. So what can we do? To analyze this, we will define new, related functions which preserve the summation, but allow for convergence.