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.