All I knew from my student mathematical education about Euler's constant, which is conventionally denoted γ, was that it exists, and that it is not even known if it is irrational. That is to say, I knew little more than that
exists. Until reading Havil's book I'd no idea what joys I had missed. For example, and this is only among the easier results, the harmonic number
is never an integer, and if n is not 1, 2 or 6 it is not even a terminating decimal.
This book contains clear explanations of a number of important facts about γ and a number of interesting sidelines. One proof of the infinitude of primes which is given here is that the harmonic series restricted to the primes,
1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ...
itself diverges (and it too is never an integer). The convergence of the expression defining γ is slow, and Havil discusses ways in which faster approximations have been found, bringing in Euler-Maclaurin summation and the Bernoulli numbers. The connection to the (Riemann) zeta function is also presented, and we are taken as far as the prime number theorem. At every stage the explanations are lucid and the history of mathematics pertinent.
Havil also describes a number of ways in which γ occurs naturally. Some of these are straight-forward enough: the number of record-breaking years in a run of n years of weather should be about Hn, on elementary statistical grounds, and so apparently it is. But γ also occurs in the famous problem of choosing the best from a sequence of examples. The examples come by one after another, you wish to choose the best: how do you do it? The answer is to reject the first r options on the list, then choose the first one that comes along which is better than your best reject, where r is about n/e. An even more satisfying occurrence, well described by Havil, is the strange matter known as Benford's law: the fraction of numbers whose first significant digit is d is log10 (1 + 1/d). Havil shows that Benford's law is a consequence of scale invariance.
The book is enjoyable for many reasons. Here are just two. First, the explanations are not only complete, but they have the right amount of generality. They are described in such a way that nothing comes up as a dreadful trick, but rather as an illustration of a general way of thinking which could reasonably be expected to pay off in the appropriate circumstances. There is an irreducible amount of dexterity and cunning needed in what is, after all, analytic number theory, and that's no bad thing, but the author never forgets that the reader needs to be persuaded at every stage. Second, the pleasure Havil has in contemplating this material is infectious. This comes out in his choice of material, in his commentaries, and in the unexpected extras. For example: Benford's law provides a statistical check against fraud, and has been used to effect in the courts.
Jeremy Gray ( firstname.lastname@example.org
) works at the Centre for the History of the Mathematical Sciences of the Open University. He also teaches one term a year at the University of Warwick and is an Affiliated Research Scholar at the Department of History and Philosophy of Science of the University of Cambridge, England. He works on the history of mathematics in the 19th and 20th Centuries, with a particular interest in complex function theory and geometry.