OK first the review – go read the book – Prime Obsessionby John Derbyshire. ’nuff said. That is all, there is nothing else to say about this book – if you want to know more about the Riemann Hypothesis, this is the book to read. And if you want to know what is the Riemann Hypotheis too, this is the book to read. And knowing both is equally important, if you are fascinated by prime numbers (and numbers in general too). There might be others (and I haven’t read them yet !) but this book, is a definite read. If there is any complaint I have about this book, it is that sometimes the author does the wave of hands to say trust me this is the case, which I guess you need to, because one can’t explain all the proofs in mathematics. And trying to explain the zeta function, well, that is a different challenge altogether.
You need to read this book for three reasons
- To know the background for one of the Millenium problems (and one of the problems in Hilbert’s 23 problems [it is the number 8])
- To know what are the approaches mathematicians are taking to tackle the problem (quantum physics anyone?)
- And more importantly, to try and understand Riemann’s hypothesis itself, and how does knowing whether the non-trivial zeroes of the zeta function having a real part of 1/2 help in determining the distribution of the prime numbers
The book is divided into two sections. Section 1 deals with The Prime Number Theorem (PNT) and the second section deals with Riemann Hypothesis. Apart from these two sections, the author categorizes the even numbered chapters on the history of the development leading to the hypotheis (and to the future too) and the odd chapters are mathematical expositions.
The Prime Number Theorem
A relatively less known fact is that there is already an equation which tells us the number of primes that are available less than a given N. The equation says, very simply π (x) ~ N / log N (the π is the prime counting function, which is used to define the number of primes less than a given N. And this was determined in the 18th century by Gauss (and others). An implication of the PNT is that the probability of a number N to be prime is ~ 1/log(N) and that the Nth prime number is NlogN (all logarithms to the base e). So, if this was known so long ago, then what is the big fuss about the hypothesis ? Hmm, well the problem is the little squiggly and that is what (in a round about way) is what Riemann was hypothesizing about.
I won’t go into the details of how the PNT and the Riemann Hypothesis are related, but in this section, I saw the most beautiful mathematical equation that I encountered till now. Euler’s identity might be considered the most beautiful mathematical proof by most of the people, but for me this equation and its proof gave enough evidence to what beauty in Mathematics is all about. This proof, what the author calls The Golden Key is the Euler product formula. The proof in the book is around 2 pages and when I reached the final sentence, I was, for the lack of a better word, overwhelmed ! And look at that equation, it is so simple Σ n -s = Π(1-p-s)-1. This basically says, the sum of the numbers till n is equal to the product of primes. Sum of numbers equals the product of primes (yes I am paraphrasing it for effect!). But think about it, it is just awesome ! And this is also important because, the LHS is actually the Zeta function and the RHS is the product of primes.
The author goes further to explain how the PNT is improved further (enter calculus, err you thought this was about numbers, and so did I) where, π(x) ~ Li(x) where Li is the log integral function. There are lots of these proofs that the author describes, all leading up to the crescendo, which he calls the Turning the Golden Key, which is the chapter 19.
In the second section, the author tries to explain the hypothesis and the higher math behind it. I must there were parts in this section which were OHT for me :(. For example, in the chapter 13, the author tries to explain the complex plane and how the function’s graph looks (knowing how a function’s graph looks definitely provides a lot of insight into how the function behaves), but I couldn’t really grasp it very well. And further in the chapter 20, he talks about Reimann operators and other approaches, which was too complex for my rather limited mathematical knowledge (no pun intended). For me, the section 2 is something that I will have to re-read till I get my head around it. For people who have a better grounding in mathematics than I do, I guess it will be easier
For me the section 1 is the favorite. The mathematics in that section is very accessible, understandable and the author does a splendid job in explaining the required mathematics. I might have read a section once or twice to understand the implications, but for most part – it was easy to follow.
And the other part of the book – the history of the developments. This is, lets just say, fabulous ! I could just read the historical chapters alone (yeah, yeah, because I don’t understand the Math so much, ok, go away !) to realize the amount of hard work involved in solving this equation. Every mathematician I knew of (and most that I didn’t know of) figured in the search for the proof of this problem. And yes, this is a problem though so simple sounding, is so complex. So, what was the problem statement anyway, you ask. Oh well, let’s see, do the prime numbers have a pattern ? Can you find out what is the next prime ? While you try to figure that out, pick up this book and read it, you will not regret it. A perfect 10 for this book. I am going to treasure this book and read it more than once. And looks like there are other books available on the Zeta function.