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Prime Obsession
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.
Ummm...what would its peers be? Just how many "classic" math books does the lay-person have now?
Could it be that the lay-person wouldn't be interested in any book about math, no matter how well written?
I dunnnoooo...almost sounds completely probable.
Why is it, that if you have studied math that people get you these books for Christmas, etc. People say, "Wow, he's into math, I'm sure he'd like that", when books like this are written for the lay person, as a fun introduction to the subject. People don't get Literature majors "Shakespeare for Dummies".
Rhymes that keep their secrets will unfold behind the clouds.There upon the rainbow is the answer to a neverending story
One man's propaganda is another man's editorial opinion.
Why must we use such slanted terms to describe the views of people we disagree with?
perhaps I just answered my own question.
There's a Mercedes gap too. I want one and can't afford one, but it's not government's job to do anything about it.
It's called white space. Look into it. Humans parse on it much faster then they parse on operators.
I currently have no clever signature witicism to add here.
I second this - Mathematics for the Million is truly a classic that belongs in this category that the story author referred to. It should be handed out to every child at a reasonable age, so that they can use it as an extra-curricular guide as they learn their way through the horrors of school-taught math.
11*43+456^2
After working at Initech for a year and not using any of my math skills, this was a welcome dip in the math kiddie pool.
I would probably need to do a few laps before I could go playing around near the high-dive again or anything. I don't think this speaks to my grasp of the subject or my intelligence, but to my complete abandonment of study for a long period.
This wouldn't be a book to get someone that works in a heavily mathematical field, but its a great choice for the coder in your live that likes math but has to write boring code all day to pay the bills.
Not to forget N. Bourbaki, "Elements de Mathematique".
I'm in charge in Moria!!
Don't blame Durga. I voted for Centauri.
People get those gifts because they try. They don't understand math at all, but they know that you do "something mathy".
Exactly right. They are trying to get you something that they think you might like even though they don't know very much about math. Instead of the grandparent getting all hot under the collar that his family and friends dare insult his grant intellect by purchasing him a "Math for Dummies" book (as he seems to think this historical work is), he should feel gratified to know that he family cares enough about him to actually put forth some effort to getting something that attempts to match his interests. There are lots of people who simply buy generic gifts for family like socks or shit like that. Isn't this book a lot better than a gift like that?
Reading the grandparents rant, I was reminded of an article in The Onion awhile back about some film snob getting all upset because his family -- damn their incompetance! -- dared buy him the widescreen edition of one of the Matrix sequels when he actually wanted the letterboxed edition (opportunity for karma whoring here if someone can link to it). For chrissake, your family and friends are trying their goddamned best and you get your panties in a bunch over details? That's so incredibly childlike, I can hardly believe this above "rant". Christmas isn't really about getting exactly what you want -- at least once you're an adult it's not. Christmas is just an opportunity to get together with loved ones and exchange gifts as a token of affection. It doesn't have to be the "perfect gift"; as long as it's somewhere in the ballpark you should feel happy that your family is at least aware of your interests.
GMD
watch this