Gamma: Exploring Euler's Constant [Hardcover]

Gamma is different. While you can understand the theory presented in Julian Havil's book if you stayed awake during second semester calculus, you definitely have to work at it. The requisite analytic number theory presented may turn away the average reader if they are not prepared to make the commitment to stay on the roller coaster for the full ride.

You will be rewarded if you can break through the initial 2 or 3 chapters introducing us to the logarithm and the harmonic series. To be fair, as a previous reviewer has noted, the material on Napier and the logarithm has been done in a more satisfactory manner by Eli Maor in his book on e. But this is only a minor drawback. As long as you are comfortable with the natural logarithm, you can omit Chapter 1 with no loss.

Chapter 4 starts off with the zeta function, arguably the most enticing and mysterious function in all of mathematics, despite approximately 150 years of analysis by the world's best mathematicians. This one function alone could arguably be said to be the genesis of analytic number theory (even though Dirichlet's work on primes in arithmetic progressions has typically been given credit for that role). All the familiar material is presented, including Euler's product formula, the "trivial" divisors of the zeta function, the infinitude of primes, Euler's evaluation of the zeta function for positive even integer powers, etc.

Of course, the gamma function makes its obligatory appearance. After having read Nahin's book on i, I was initiated into the math connecting the gamma and zeta functions. But Nahin of course could not use Euler-Maclaurin summation or the familiar inequality arguments as this would have taken him too far afield. After having read the traditional fare, such as Hardy-Wright, Apostol, Hua, et al., it was nice to see a more conversational approach to the material. I literally felt like I was sitting in Havil's office while he dissected the material for me, on a level I could comprehend.

My last comments on this book are the extras. As expected, Riemann's hypothesis and complex analysis make extended appearances. I appreciated the fact the Havil resisted the temptation to take the Riemann Hypothesis beyond the traditional mathematical lore and float off into the ethereal. This happened with John Derbyshire's otherwise excellent book "Prime Obsession", which devoted a little too much time to the psychoanalysis of Riemann, who after all, only scratched the surface of this problem. Derbyshire's book is highly recommended though for more material on the Prime Number Theorem, and some of its uses to formulate modern permutations of the Riemann Hypothesis.

He presents the usual anecdotes on Riemann and Hardy (who had a major love affair with the Riemann Hypothesis), but these are sidelines only, as they should be. Also, the material on residue integration and analytic continuation in the appendices is enormously helpful to understand the post Riemann attacks on the problem. In addition, well, it's just pretty mathematics.

The introduction by Freeman Dyson is quite impressive. How many books of popular mathematics get endorsements like that from world-class physicists? The praise is well deserved. This book belongs on every math enthusiast's bookshelf!

My one complaint about the book--and the reason for giving it four stars instead of five--is that there are times when the formulae and notation get so dense that it's extremely difficult to follow the author's train of thought: I can think of a number of places where diagrams would have helped immensely. Likewise, since there's no list of symbols or formulae, it's not a book that you can simply browse through, in the sense that you can browse through, say, "A Brief History of Time."

Finally, let me reiterate that this book assumes that you already know a fair amount of math: if you don't know what a capital pi means, for example, you're probably going to have a hard time understanding this book. But if you *do* know what that symbol means, though, then by all means, give this book a try.