The premise of this amusing crime caper is a gang of experts who pull of a successful theft of a $310 million Powerball lottery jackpot by generating a winning ticket just after the numbers have been drawn. One of the masterminds is a French mathematician (the fictional equivalent of the real mathematician Louis de Branges), who claims to have solved the Riemann hypothesis about the distribution of prime numbers. Their scheme relies on his knowledge of the gaps between prime numbers to slow down the computer as it is validating lottery slips. The story has just enough math to be tantalizing but doesn't require the reader to have any real math knowledge to understand or enjoy the book.
The quote above is from half of the writing team of "K" and "T" Reid. (Even though she contacted me by e-mail, I'm not certain if she wants her name revealed at this point. So, to be on the safe side, I've kept it somewhat anonymous.)
Although I have not read the entire book,
I can comment a bit on its presentation of the math because (at least presently), the first chapter of the book is available as a free download from their website: www.powerball310.com.
It is probably true that
"the story does not require the reader to have any real math knowledge" (quoted from the author's description above).
In fact, I suspect that once you get past a brief explanation of how the RH is supposed to allow the scheme to work, the rest of the novel will focus on exactly how the scheme plays out rather than on any mathematics.
Still, since the focus of the website is on the mathematics, I would like to just briefly register the following "complaints":
- It is nice to see this novel mention the Riemann Hypothesis (RH) -- an important and deep piece of modern mathematics. This is a conjecture about where the zeros of a certain complex function are located. That function can be used to prove a theorem about the asymptotic behavior of another function that counts the number of primes less than a given integer, but RH itself does quite tell us how the primes are distributed. For the sake of those who may not know much about it but are interested in some of the mathematical details, let me briefly outline the key points:
Berhard Riemann found a formula that can be written in lots of interesting ways. The same formula can be written as an integral with exponential functions, as a product involving all of the prime numbers, or as a sum involving a function that counts all of the primes less than a given integer. The point being that if we understood this function well we could learn about prime numbers from it because despite the fact that we can write down these formulas we don't know all of the primes or the number of primes less than a given number!
The Riemann Hypothesis is just an unproven conjecture Riemann had about where the zeroes of this function (like x=+1,-1 are the zeroes of p(x)=x2-1) are located. His conjecture is that all of the non-obvious zeroes lie in a certain location ("on the critical line") and if they do not then his conjecture is wrong. What, if anything, would knowing which is true tell us about the distribution of prime numbers? Well, there is a well-known formula that returns a good approximation to that function which counts all of the primes less than a given integer. If RH is true, then a consequence would be that this approximation is even better than we previously knew. But, regardless of whether it is true or false, it does not directly give us a better way to approximate it and certainly does not straight out give us total knowledge of the distribution of primes.
On the other hand, depending on how it is proved, such a consequence (us learning something new about how the primes are distributed) is truly possible. When I was working at MSRI, we were following approaches to proving the RH that related to mathematical physics. In particular, when one looks at large matrices filled with random numbers, a bunch of startling coincidences occur. The properties of these matrices begin to look strangely like those found by mathematical physicists studying waves and particles (as I do) and also like the patterns of prime numbers found by number theorists. This leads to the hope of proving RH and getting a better understanding of the prime numbers by thinking of them as possible energy levels of a physical system. Obviously, this has not been done yet, but if it was it could not only prove the hypothesis but give us a new insight into the distribution or identification of prime numbers among all positive integers.
Reid's description above suggests that the mathematician character in the story is partly inspired by Louis de Branges. This real (as opposed to fictional) mathematician has claimed to have proved RH, and that "proof" has not generated as much interest as he would like. de Branges is known to have claimed to have proofs of important results before, and in some cases he has proven to be correct, but the cases in which he has not have left people somewhat skeptical. More importantly, experts have argued that an older paper contains examples which refute some of the key steps in de Branges' supposed proof. I must admit that I have not checked myself and do not have sufficient expertise in this area to judge in any case. He may well be right, and there may be surprising consequences of his proof that only he knows at this point. It is easy to imagine a person in that situation deciding to use his discovery for nefarious purposes, like the character in this book. So, I guess I do like that aspect of this story.
Anyway, to summarize, this may be a great crime novel with exciting twists and turns (I can't comment having not read the whole thing), and I do like the inclusion of a character with a proof of RH that is not appreciated by the mathematics community. The description of the mathematics itself, though not entirely accurate, is close enough that it doesn't bother me much. But the thing that really bothers me is that its connection to the plot, this idea that it has something to do with lotteries, makes no sense at all. It makes me think of the episodes of the TV show "Star Trek" where they seem to be able to explain any bizzare phenomenon away by mentioning "tachyons" and "singularities in the fabric of space time". It becomes clear that whether this makes any sense is irrelevant since the whole point is just to justify some unlikely scenario. If this will not bother you, then do check out the book and come back here to let us know how you like it.
To begin with, I would not characterize the Riemann Hypthesis (RH) as "extremely arcane". Not only is it one of the most famous open problems in mathematics, but the attempts to prove it involve cutting edge mathematical techniques from just about every area of mathematics. More importantly, if the conjecture is ever proved (or disproved), everyone will of course be interested to see whether the method has implications in cryptography, an important application of number theory.
- The book says "the zeta function is Riemann's name for the fudge factor that predicts exactly how many primes there are less than a certain number". It sounds to me as if this is saying that the zeta itself gives the number of primes less than its input. There is such a funtion frequently mentioned by number theorists: it is called π(n). So, for instance, π(10)=4 because 2, 3, 5 and 7 are the four primes less than 10. However, the zeta function has the value ζ(10)=π10/93555. (Here π=3.14... is the usual constant related to the geometry of circles!) Of course, the authors are correct that there is a connection between π(n) and ζ(z), but I would not call it a "fudge factor". Rather, the connection is that it is possible to write the zeta function exactly in terms of π(n).
- Perhaps this is not actually a complaint, but I would like to re-emphasize that a proof of RH would not immediately tell us anything useful about the distribution of primes. In fact, since most people believe the Riemann Hypothesis is true, and we know exactly what it says, it would be possible to just assume it and follow its consequences even without proving it. However, unexpected consequences (such as schemes for winning the lottery or -- more likely -- reading encrypted messages) could possibly be consequences of the particular method of the proof, as I explained in the boxed description above. And so, in this sense, the book is correct that a mathematician who proved the RH might be in possession of some knowledge abour prime numbers that would allow him/her to do something unexpected.
- I don't see why a machine checking for lottery winners would care at all about the distribution of prime numbers, let alone whether it would be possible to slow it down by knowing it yourself. More importantly, although we do not know all of the prime numbers, we do know all of the ones that you can find on a lottery card, and so I don't see how the truth or falsehood of the Riemann Hypothesis would be needed in any case.