
Karl Sabbagh
Dr. Riemann's Zeros
In particular the book has quite a bit on the work of Louis de Branges who proved the Bieberbach, and claims to have a proof of the Riemann Hypothesis  but other mathematicians aren't so sure. Sabbagh explains the problems which arise in the checking of such a proof.
The book has a bit of mathematics, but is aimed at those with very little mathematical experience  I think that those with a good grasp of school mathematics who want to get a glimpse into the work done on the hypothesis might do better with a different book. But if you want an easy to read overview of the different ways mathematicians have approached such problems then you should take a look at this book.
Note: in the USA this book is entitled The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics