NTIC Long-Term Function Behavior

Chapter 20 Long-Term Function Behavior

We will now move on to think of these same functions in a different way from the previous chapter. We will examine different limits in number theory, and how integrals and calculus are inextricably bound up with this sort of question.

If, after this chapter, you are interested in more of this kind of material, definitely check out¹ Stopple's excellent [E.4.5], to which I am indebted for many of the ideas here, or the more challenging book [E.4.6] by Apostol.

Two other books with useful presentations are the terse one in [E.2.9] and the more intuitive, if shorter, one in [E.2.11]. [E.2.8, Section 3.8] has a deep but idiosyncratic presentation, as evidenced by its starting with what we give as Proposition 24.6.7!

Finally, note that some proficiency in calculus is helpful in understanding the results in this chapter, though a proper course is not necessarily a prerequisite.

Summary: Long-Term Function Behavior

Here, we investigate – and prove – what the long-term behavior of several important functions is.

The first section reviews our computation of the sum-of-squares function from the point of view of error, including the important concept of Big Oh.
In Section 20.2 we begin examining the \(\tau\) function from this perspective, though without conclusive results.
In Section 20.3 we then carefully use geometry and limits to show that the average value grows logarithmically, and can even give fairly accurate information about the error.
Section 20.4 does the same thing, but now for the sum of divisors function.
Finally, Section 20.5 gives a short summary and then asks (but does not answer) the same questions for \(\phi\text{.}\)

The Exercises focus mainly on understanding Landau notation and filling in details of the proofs.