γ may be the most important mathematical constant after e and π, but its definition is rather less straightforward: it is the limit for large n of the difference between the harmonic series 1 + 1/2 + 1/3 + ... + 1/n and the natural logarithm ln n.

Havil begins Gamma with a twenty page chapter on Napier and the early history of logarithms. This is hard to follow in places because it takes a historical approach; it is also not representative of the book in that subsequent chapters, while they do incorporate history, don't do so in this detail.

The core of the work introduces the harmonic and sub-harmonic (omitting some terms) series and zeta functions, before looking at the gamma function (generalising the factorial), its historical origins, and Euler's Identity, which links gamma via the zeta functions to the prime numbers. It then explores what gamma is, approached as a decimal and as a fraction, and some of the places it appears, in connection with the alternating harmonic series, in analysis, and in number theory.

Many of the results here are startling but at the same time baffling: "Other, nameless integrals and limits involving γ are easy to find." They are often produced by conceptually opaque proofs, involving clever algebra and integration tricks (there's a lot of integration by parts), but not introducing any new ideas or leading anywhere.

More interesting in many ways are the applications. Havil presents a multitude of applications of both the harmonic series and of logarithms, which involve music, record-setting, testing, overhangs, information theory, Benfords Law (on the frequency of first digits) and continued fractions, among other topics.

He ends by returning to the distribution of primes and describing the various ideas leading up to the Riemman Hypothesis. This is quite difficult and brings in some complex analysis (which is explained in thirty pages of appendices).

Even without that, Gamma is not aimed at a general audience. It includes quite a lot on history and applications, but it is at heart still a presentation of mathematics. And, while it doesn't try to be fully rigorous and it has a pleasantly informal style, avoiding lemma-theorem stodge, it is quite dense, both visually and conceptually. It assumes familiarity with sequences and series, complex numbers, and limits and calculus, probably at undergraduate level, though someone with solid high school mathematics should be able to cope.

June 2015

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