1 March, 2005Issue 4.2EuropeScience

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Prime Obsession

Florian Huehne

John Derbyshire
Prime Obsession: Bernhard Riemann and the Greatest Unsolved Puzzle in Mathematics
Joseph Henry Press, 2003
448 pages
ISBN 0309085497

Mathematics and mathematicians have featured prominently in Hollywood in recent years. However, behind the veneer of scrawled equations, whose logical flaws everyone with a half-knowledge of mathematics will easily detect, there is little to discover. Films such as Good Will Hunting, Pi or A Beautiful Mind stay well within the boundaries of their respective genres and are more occupied with endorsing familiar clichés about mathematicians than with scientific content.

On the other end of the spectrum, there has lately been a more serious rise in public interest in mathematics. This might have been sparked by recent advances: only in 1994 did Andrew Weil prove Fermat’s Last Theorem, the famous conjecture that there are no whole-number solutions to the equation xn + yn = zn (when ‘n’ is a whole positive number greater than two). The proof of this theorem took the academic community 357 years to achieve, spawned entire new areas of mathematics and made Weil the superstar of mathematics when he published his final result in 1994. Other equally difficult problems have received similarly widespread attention: the Four Colour Theorem (that four colours are sufficient to colour any map in the plane, no two adjacent regions having the same colour), which was proved in 1976 using no fewer than 1,200 hours of computer time; and Goldbach’s conjecture (every whole number greater than two is the sum of two primes), which remains unproven. Such recent high-profile advances have subsequently inspired a new generation of authors who are trying to do what mathematicians are notoriously poor at doing: describing complex matters in simple terms.

There are now popular books on Abel’s theorem, on Euler’s constant gamma and several on the Riemann Hypothesis. Of these, the lattermost presents the most significant challenge for the popular maths writer. The Riemann Hypothesis is a conjecture stating that in all the interesting places where the zeta-function is zero, its argument has a real component of 1/2. As a theory, it is not only difficult to state in simple terms, but has proven to be more elusive to solve than many long-standing problems believed to be of similar difficulty. Over the years the proof of the hypothesis has become an end in itself despite the fact that its proof would have practical repercussions in mathematics, physics and cryptography.

Fortunately, John Derbyshire belongs to the breed of authors capable of putting complicated matters into simple words without overly diluting his subject matter. A mathematician and linguist by training, he has come to some fame as the author of Seeing Calvin Coolidge in a Dream, a 1996 novel about a Chinese immigrant coming to grips with American culture. Prime Obsession, written in a style wavering between that of introductory computer programming books and the Encyclopedia Britannica, marks a clear thematic and stylistic departure from his earlier work.

The chapters of the book alternate between mathematical expositions and broad historical accounts. While this may invite the reader who is not interested in mathematical intricacies to skip the technical chapters, this is by no means recommended: not only does the line between purely technical and historical accounts blur, especially in later chapters, but the reader will only gain a full appreciation of the historical facts with an understanding of the basic mathematical results that are underpinning the historical development. The book is aimed at a ‘curious but nonmathematical’ audience; however, my feeling is that some of the later chapters might be a bit too difficult to grasp for a reader only fulfilling these minimum requirements. Moreover, the author’s claim, that the Riemann Hypothesis cannot be explained using mathematics more elementary than used in his book, could potentially face some opposition from authors such as Marcus de Sautoy and Karl Sabbagh who have recently published books on the Riemann Hypothesis that seem to get by with almost no mathematics at all.

The thematic centre of Derbyshire’s exploration is Riemann’s 1859 paper, ‘On the number of prime numbers less than a given number’, which he presented at the Berlin Academy at the age of 33. Throughout the book, Derbyshire refers to the results of this paper and to the hypothesis presented therein, but it is not until the penultimate chapter, however, that the reader gets an idea of the full consequences of Riemann’s work.

The book begins with an introduction to analysis, the discipline of mathematical thought that is concerned with the study of limits particularly of infinite sequences. The author then delineates some ideas of number theory, in particular the prime number theorem (PNT) which gives an estimate of how many prime numbers exist with a value lesser than a given number N, assuming that N is a large number. While the reader is now familiar with the two strands of mathematics that Riemann successfully merged in his famous paper—analysis, and number theory (i.e. the study of the properties of integers) —it remains unclear until much later in the book, how the two interact. The technical chapters that follow are devoted to the analysis of the zeta function using the simple analytic tools that were given in the beginning of the book. Riemann took the zeta function, which had been studied by many previous mathematicians, and showed how to think of it as a complex function. This extended, ‘complex’ zeta function, referred to as the Riemann zeta function, takes the value zero at even negative numbers (the so-called trivial zeros). Riemann’s hypothesis states that it also takes zeros on the critical line, a line of complex numbers with real part 1/2.

In order to explain these concepts in greater depth, Derbyshire then introduces basic ideas of function theory and algebraic concepts such as fields. Mercifully for the lay reader, he diverts from this technical exposition to discuss some results on the Riemann hypothesis in the realm of quantum physics and operator theory. Only in the last two chapters, however, does he finally establish the connection between the prime number theorem, which states that the number of prime numbers less than any number x is accurately approximated by x/ln x, and the zeta function. Thus equipped, the reader understands for the first time why the Riemann hypothesis is of theoretical importance in number theory specifically and in mathematics as a whole. The last two chapters are mathematically much more demanding than the preceding ones, which might explain why Derbyshire has been holding them back for so long. While intensive, however, in the end these technical chapters give an accessible and interesting overview of the theory underlying the Riemann Hypothesis.

Although the thematic and chronological centre of the book is Riemann’s 1859 paper, the historical horizon stretches far beyond the nineteenth century. In an order that is neither chronological nor terribly thematic, Derbyshire zigzags through centuries of mathematical thought, employing a recipe that too often seems to consist of overly familiar ingredients. The main theme here is the nutty professor cliché supported by a rosary of anecdotes. He quotes, for example, from a list of New Year wishes of G.H. Hardy, a British mathematician active at the beginning of the last century, which (naturally) includes proving the Riemann Hypothesis, but which also includes disproving the existence of God and murdering Mussolini. While these anecdotes spice up a text which could otherwise seem over-infused with mathematical intricacies, they also seem to be too obvious and self-consciously inserted to effectively strike a balance between the technical expositions and historical accounts. Indeed, more often than not the reader gets the impression that the sole raison d’être for these historical interludes is to make the whole text more digestible and, possibly, more marketable.

Prime Obsession would have greatly benefitted from a tighter integration of the non-technical sections and mathematics as the alternation of technical and historical chapters often seems artificial and at times impedes the flow of the narrative. On the other hand, it is exactly one of the strengths of Derbyshire’s book to bring mathematics into the historical context, and on the whole, he might be forgiven for sticking a bit too closely to his recipe. Certainly, the reason why Derbyshire’s book does ultimately prove valuable is precisely because of his convincing integration of challenging mathematic theory with its cultural context, despite his failure to balance them quite as effectively and gracefully as he might.

Florian Huehne is a German DPhil student in Maths (with an emphasis on financial maths) at Exeter College, Oxford.