Prime Obsession

jkauzlar writes "Popular mathematics books don't come along often and when they do, they're only occasionally worth the read. John Derbyshire, a controversy-stirring political propagandist by day, and mathematician-enthusiast by night, has composed what may turn to out to be one of the classics of mathematical literature for the lay-person." Read on for the rest of jkauzlar's review. Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics author John Derbyshire pages 422 publisher Plume rating 9/10 reviewer jkauzlar ISBN 0452285259 summary History of the attempt to prove the Riemann Hypothesis

Bernhard Riemann came to the University of Goettingen in 1846 at the age of 19, originally to study theology. The University, however, was home to Carl Friedrich Gauss, "the greatest mathematician of his age and possibly of any age," and the impressionable young Riemann, succumbing to the privilege of Gauss's presence and following his already blossoming interest in mathematics, refocused his studies on the area in which he would soon attain distinct immortality. As early as 1851 he was impressing even Gauss with the results of his doctoral dissertation and in 1859 was appointed a corresponding member of the Berlin Academy. To this honor, Riemann responded with his most famous paper, entitled "On the number of prime numbers less than a given quantity," containing therein what became known as the Riemann Hypothesis.

At the heart of the RH is the Zeta function which, in its basic form, looks like this: Z(s)=1 + 1/2^s + 1/3^s + 1/4^s + ... and which, through some simple algebraic manipulation as demonstrated by the mathematically gifted journalist Derbyshire, can be given in the form (1 - 2^-s)^-1 * (1 - 3^-s)^-1 * (1 - 5^-s)^-1 * (1 - 7^-s)^-1 * ... And it is in this second form which Derbyshire calls "The Golden Key" where the non-mathematician gets the first glimpse of the Zeta function's relationship with prime numbers.

But where this Golden Key appears as this "novel's" turning point--its central conflict-- it is not until Prime Obsession's climax when the Key is at last turned and the Zeta function's true relationship to the prime counting function pi(x)--the number of primes less than a given x--is at last made clear. Along the way, from the introduction of the Zeta function to the final explanation of its relevance to prime numbers (the turning of the Key), Derbyshire enlightens us with clear, mostly English language descriptions of the mathematics involved, as well as plentiful anecdotes that give readers a sense of the life and work of the major figures in the history surrounding the RH from Euler, Gauss and Dedekind in the late 18th century through Riemann's 1859 paper, and from 1859 onward to recent advancements in the '80s and '90s.

The Riemann Hypothesis states that "all nontrivial zeros of the Zeta function have real part one-half." Understanding the statement of the hypothesis is Derbyshire's first mission for the reader. In short, most functions with a dependent variable, say f(x)=x^2-2x+1, have a value for which if you replace x with this value, the function returns zero. In the example given, it is at the value x=1 where f(x)=0. The Zeta function has an infinite number of these zeroes and an infinite number of these is "non-trivial." The non-trivial zeroes come from complex number values. Riemann's guess, his hypothesis, is that the real part of each of these non-trivial zeroes is equal to one-half. The imaginary part can be anything.

Derbyshire explains all of the mathematics in very readable language. It's unlikely that anyone who did well in high school mathematics will not be able to follow Derbyshire's mathematics (and it's unlikely that those who didn't do well will pick up a 400-page book on this topic). The Zeta function is explored from a number of angles--numerically, graphically, algebraically, statistically, and there's even a link between the non-trivial zeroes of the Zeta function and quantum physics! By a larger margin, however, Prime Obsession's intrigue lies in Derbyshire's expositions on Riemann, Hilbert, Turing, Gauss, et al, as well as those modern mathematicians he's interviewed personally. The line between the mathematical half of the book and the historical is clearly defined; the odd-numbered chapters are devoted to the former, the even to the latter.

Those fans and foes of Derbyshire's most public line of work as a journalist/editorial writer for National Review will be comforted to know all political polemics have been set aside. John Derbyshire gives a virtuoso performance as an informed journalist and maintains his stance as a personable and careful guide through a sometimes difficult terrain. Anyone with some interest in the topic will find it hard to put down Derbyshire's book once begun. If we are lucky (hint, hint, JD) perhaps Derbyshire's next book will cover the newly-proven Poincare Conjecture ...

You can purchase Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics from bn.com. Slashdot welcomes readers' book reviews -- to see your own review here, carefully read the book review guidelines, then visit the submission page.

Comments

If you are interested in solving math puzzles

[suso]

You might check out my current MD5hash Challenge. Some people have told me that it is impossible to solve, some have said that mathematically it is solveable.

Not quite related to primes, but close and can certainly create an obsession. Also, look behind the scenes for something simpler to solve.

My favorite.

[standards]

My favorite book on math is The Mathematical Tourist by Ivars Peterson.

It's very readable, and has chapters on interesting stuff like knot theory, cellular automata and primes.

I highly recommend it. It isn't going to turn anyone into a math professor, but it is very interesting reading.

Why I believe this book to be of interest.

[stromthurman]

I know very few mathematicians and math students who aren't familiar with the Riemann Hypothesis (largely due to the million dollar prize associated with its proof), so a book exclusively on such a topic probably wouldn't interest too many people. What makes this book interesting, at least to me, is the Math History covered in it. In particular, the author goes into great depth into the personality and character of each of the principle figures in this book: the anecdote regarding Hilbert's torn pants, Gauss's (perhaps justified) arrogance, and Riemann's quiet nature. All of these aspects of the book add a lot more depth to the people behind this problem, and I find that to be far more valuable, as a mathematician, than yet another essay on the Riemann Hypothesis.

I agree with the reviewer's sentiment that the book is well written, and it is very enjoyable. The author writes in a very audience-centric fashion, even going as far to discuss the "scaffolding" of the book itself (all of the "hard math" stuff is found in odd chapters, the author had debated putting this information in only the "prime" chapters, but then said "there is such a thing as being too cute.")

Anywho, if you have a math friend you need to buy a gift for, definitely consider this book.

Re:lay person?

[Mr. Flibble]

Hey buddy - got six quarters for a dollar?
Consider yourself "shown".

I am not speaking of general math - rather I am speaking of the esoteric stuff such as "new math" stuff that has no "purpose" other than to be a neat trick.

I was deeply impressed by Richard Feynmans chapter on his reviewing high school math books. He was livid that a number of things being taught were useless. He wanted the books to teach the students not only what they were learning, but why. One example has him in an uproar because there was a question about taking the average tempurature of a number of stars. This made Feynman angry because there is no reason to get an average star tempurature for a number of stars, it is just not something that you do. Feynman called it "a trick to get the students to add".

Furthermore, he was furious at a physics problem in one book, that had wrong answers, and in fact, Feynman actually performed the experiment listed in the book, and found out the "observed" results were wrong. The author did not even take the time to DO the experiment listed.

Again, this made him furious because he felt that teaching students math in a deceptive manner would never give them a feeling as to where the math can take you in fields in the sciences. I agree.

So, I don't want to learn fluff. I was at a disadvantage because I was just told "learn this" and in answer to the question of "why?" I was only given "so you can pass the exams."

In high school I deeply wanted the answers to some questions in Physics, that were available with mathematics, but I was not shown these, and I developed an unfortunate disgust with mathematics because of this.

So many people here on slashdot can take me to task for being bad at math - and I know I am. I don't know if you would have been so interested in it either if it was drilled into you in a dull manner, and a feeling that it lacked a purpose.

Am I learning math now? Yes, but then I understand much more about the why, the how, and the history now than I did then. I don't know about the rest of you but I detest rote learning. So take me to task on my math skills if you wish (or my typing :) ) but I can see enough in myself that I want to change, and I am making the effort. Not all people can say the same of themselves.