After Cambridge mathematician Andrew Martin proves the Riemann Hypothesis, he is replaced by an alien whose job it is to prevent news of the discovery from spreading as it is their belief that humans are not yet ready for the power it would afford them. The alien doppelganger surprises Martin's family and colleagues by being seemingly more human than the emotionless mathematician ever was, and they surprise him by being more worthy than the primitive creatures he had been led to expect.
Of course, there is no reason to think that a proof of the Riemann Hypothesis would actually have any dramatic impact on the human race. Haig's claim that it would imply the existence of a "pattern" for the "first hundred thousand or so primes" is inaccurate, probably a misunderstanding based on the conjecture has been checked for the first hundred thousand or so primes. In fact, as you can read about in greater detail here, the Riemann Hypothesis is the conjecture that the zeroes of a certain function that can be written in terms of the prime numbers are all either negative integers or complex numbers with real part equal to 1/2. Of course, it is possible that a proof that this is true would provide knowledge that could have some important consequences that we cannot yet imagine, but I would not want anyone to misunderstand and suppose that this question, intriguing as it is to mathematicians, is necessarily of great importance to anyone else. The only immediate consequence I know of for the Riemann Hypothesis itself, which would not excite anyone but an expert number theorist, is that a certain known approximation to the function that counts the number of primes less than a given number would be known to be a slightly more accurate approximation than it might otherwise be. Hardly an application of Earthshattering consequences.
But, then, the mathematics is not really the main focus of the book. Told from the point of view of the alien who has taken over Andrew Martin's life, it is his growing appreciation of humanity and the things we have made (from peanut butter to pop music) that seems to be the key point.
