Summary: Counting and Summing Divisors
This chapter investigates the surprisingly wealth of questions arising from one of the oldest arithmetic functions.
- We first define \(\sigma(n)\) in Definition 19.1.1, and encourage a lot of exploration!
- The next section proves a number of important facts about these sums, including multiplicativity as a corollary of the quite general Theorem 19.2.7.
- Section 19.3 explores the size of the sum of divisors function.
- We next turn to a Characterization of Even Perfect Numbers. There are many interesting definitions here, and we even discuss an ancient way to Get Amicable Numbers.
- Finally, we learn not only that Theorem 19.5.2, but that no one really is sure whether they exist at all!
There is a very broad variety of Exercises looking at all the definitions, and their variations, related to summing divisors, ending with some interesting historical ones.
