Low Level Books on Conjectures/Famous Problems

Question

I am currently an undergraduate math/CS major with coursework done in Linear Algebra, Vector Calculus (that covered a significant amount of Real 1 material), Discrete Math, and about to take courses in Algebra and Real Analysis 1. I was wondering if there are any books about famous problems (Riemann, Goldbach, Collatz, etc) and progress thus far that are accessible to me. If possible, I would prefer them to have more emphasis on the math rather than the history. As an example, I enjoyed reading Fermat's Enigma by Simon Singh but was disappointed in how it glossed over most of the math. If there are any suggestions I would greatly appreciate it.

Answer 1

Jeff Lagarias edited a book of papers about the Collatz problem. Some of the papers should be quite accessible to you.

Answer 2

The Goldbach Conjecture by Yuan Wang.

From the book's description:

A detailed description of a most important unsolved mathematical problem - the Goldbach conjecture. Raised in 1742 in a letter from Goldbach to Euler, this conjecture attracted the attention of many mathematical geniuses. Several great achievements were made, but only until the 1920s. This work gives an exposition of these results and their impact on mathematics, in particular, number theory. It also presents (partly or wholly) selections from important literature, so that readers can get a full picture of the conjecture.