Sabbagh's goal is to give some insight into mathematicians' lives by focussing on one particular, and very deep problem: the 150 year old Riemann Hypothesis. He has a tough task: the problem itself is rather harder to explain than the much more accessible Fermat's Last Theorem, involving as it does an infinite sum of complex powers of complex numbers, and the mathematical approaches to its proof are all but incomprehensible to any other than professional mathematicians. Yet he succeeds very well indeed.
It takes a while for the book to get going, because in order to even explain the problem, Sabbagh has to explain complex numbers, and raising to a power, which I am fortunate enough to know about already. (There are some very good technical "toolkits" in appendices, summarising the relevant mathematics.) After about 50 pages, however, things take off, leading to lots of new, deep, and fascinating details. The hypothesis is so fundamental that every mathematician working on it seems to be using, even inventing, a whole different branch of mathematics to tackle it. I was particularly intrigued by the spectral approach and the link to physical systems -- how such a concept in the purest of pure mathematics might be related to a real world physical system is nothing short of astounding.
Sabbagh has a way of explaining extremely deep concepts in a way that gives, not so much understanding (that would be too much to ask for), but certainly some kind of feeling of what the problem is all about. He also manages to convey how very different these pure mathematicians are from the rest of us, but in a sympathetic way that makes their fascination and obsession with the problem credible. This is a very good read.
(Coincidentally, I've recently read two fun fiction books, Doxiadis's Uncle Petros and Goldbach's Conjecture, and Stephenson's Cryptonomicon, that are related to this. I had thought some of the plot details about combinations of sums in the former, and Turing's Riemann zero computer in the latter, were just fictional detail invented by the respective authors -- but apparently not!)