NTIC Counting and Summing Divisors

Chapter 19 Counting and Summing Divisors

Among all the possible arithmetic functions one could discuss, there is one family which is both truly ancient and part of cutting-edge research. We'll let ourselves be inspired by the summations in the previous chapter, by summing the simplest functions of all and seeing what we get.

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Summary: Counting and Summing Divisors

This chapter investigates the surprisingly wealth of questions arising from one of the oldest arithmetic functions.

We first define $σ (n)$ 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.