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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 pn as an indexing tool denote the set of primes which divide n=p prime pe (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.)
σ(n)=pn(pe+11p1)=pn(1+p+p2++pe)
ϕ(n)=npn(11p)
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.
σ(n)n=pn(pe+11p1)pnpe=pnp(1/pe)p1
Alternately, to avoid fractions:
σ(n)n=pn(1+p+p2++pe)pnpe=pn(1+p1+p2++pe)
Note that ϕ(n)n=pn(11p).

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.
It’s clear with σ that this is the case, since we defined (in Definition 19.1.1)
σ(n)=dnd
We can even cleverly add up the divisors in the opposite order to get the slightly more felicitous
σ(n)=dnnd=ndn1d.
This led us directly to writing σ(n)n=dn1d in Fact 19.4.9; now we can also write it as dnu(d)d.
With ϕ we have something to prove to make this connection, but not much. In Fact 23.3.2 we saw that ϕu=Nϕ=Nμ. Equivalently, we have Möbius-inverted Fact 9.5.4 to obtain, from dnϕ(d)=n, the formula
dndμ(nd)=ϕ(n)
By adding the divisors in the opposite order (alternately, by noting is commutative) we can write
ϕ(n)=μN=dnμ(d)(nd)=ndnμ(d)d,
which allows us to also write the fraction as
ϕ(n)n=dnμ(d)d.
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.