Chapter 24 Infinite Sums and Products
We are almost at the very frontiers of serious number theory research now. In order to start to understand this, we will need to introduce two final concepts:
These concepts both deeply involve infinitely applied operations, and are what this chapter is about. If you wish, think of this chapter as the ‘infinite’ version of the previous chapter on new functions.
Summary: Infinite Sums and Products
Our penultimate chapter asks what happens if we take our formulas for arithmetic functions and add infinity to the mix.
The first section, Section 24.1, examines the connection between products and sums for arithmetic functions.
Then we define the Riemann zeta function and examine some of its basic properties.
What happens more generally when we go to infinity? We get Dirichlet Series and Euler Products.
The next section examines multiplication of these infinite series and products in Theorem 24.4.3.
We then investigate how these infinite processes work with the \(\phi\) function, as well as show technical details of convergence in Fact 24.5.5.
In the final section we can now prove Four Facts of high interest, including my favorite, Proposition 24.6.2.
The Exercises begin winding down, as we give more conceptual activities.