The Riemann Hypothesis is a statement about where the zeroes of a certain function -- the zeta function -- lie. It is connected to number theory since that function can be written in an elegant way in terms of the prime numbers. (And so, perhaps our difficulty in identifying the zeroes of the function is related to our difficulty in describing which numbers on the number line are prime and which are not.) It is one of the more famous open problems in mathematics, and consequently one would receive not only fame but riches as well for resolving it!

The connection to quantum physics is very much as described in the book. It won't be easy to give a non-technical summary in greater detail, but let me try: You may well have encountered matrices of numbers at some point in your mathematical education. These are rectangular arrays, blocks if you will, of numbers. If the matrix is square (same number of rows and columns), then there are certain special numbers associated to it called its eigenvalues. (If you think of a matrix as an operator that turns vectors into other vectors by multiplication -- flippling, twisting and stretching them -- then the eigenvalues are a measure of how much it stretches those vectors that are not twisted.) As it turns out, this is a VERY important idea in quantum physics. According to the dogma of quantum mechanics, every measurement that you can make is just the eigenvalue of some associated operator. So, in particular, the energy levels of electrons that you may recall from chemistry are nothing but the "vectors" associated to the different eigenvalues of the energy operator. Now, if you're thinking that this doesn't sound very much like number theory, you're right. That's why it is such a surprise that there is a connection, even if it is only a connection of vague similarity at this point without any proof. The thing is, when Hugh Montgomery got some results about the spacings between the zeroes of the zeta function, physicist Freeman Dyson recognized it as being precisely the same sort of statistical results that one gets about the eigenvalues of the matrices used to model electron energy levels. So, one approach to proving it is to try to find an operator that really has the zeroes of the zeta function as its eigenvalues. As the book says, at this point it is only wishful thinking, but it is something that people are really pursuing.

For more information, try reading this article by Ivars Peterson.

The plot also eventually depends on the relationship between prime numbers and internet security. This is not as vague as the hypothesized connection to physics described above. In fact, modern cryptography is based on number theory and there is the very real possibility that a mathematical discovery could make it possible for someone to read all of the encrypted messages on the internet (including your credit card number when you use it at an online store)! As far as we know, we are all safe for now because nobody knows how to quickly factor really large numbers, but a discovery like the ones considered in this book could change everything. For a more detailed discussion of how number theory is used in cryptography, check out the boxed information on the review of Eye of the Beholder.

Yes, I do generally point out mathematical errors in fiction on this website. (Sometimes I receive complaints about that from people who think it is an unfair to criticize fiction for such errors, but I think it is more important for me to make sure that people are aware of these mistakes.) So, if Mr. Anonymous (or any of his friends) would care to write in with a more specific list of mathematical errors in "Case of Lies", I'd be happy to post them here!