The Riemann hypothesis

The New York Times

Karl Sabbagh is a producer and writer for the BBC, and he brings that perspective to his writing. — James Alexander

Publishers Weekly

With Fermat's Last Theorem proved, the Riemann Hypothesis has become math's most glamorous unsolved problem, and has spawned a growing literature seeking to explain it to lay readers. Unfortunately, this curious genre is overshadowed by the fact that the hypothesis itself is incomprehensible to anyone without a Ph.D. Sabbagh, author of A Rum Affair, struggles manfully with this problem, and gives impressively lucid explanations of such preliminary subjects as prime numbers, logarithms, infinite series, algebraic equations and matrices. But even with all this background, the hypothesis remains such an opaque abstraction that, at one typically baffling juncture, the author throws up his hands and instructs readers to either "sign up for a few months of complex analysis and number theory, and then pick up the book again in a year or two" or else just "take it on trust." To help elucidate the material, Sabbagh includes many lengthy excerpts from interviews with mathematicians, who, he claims, "see truths with a clarity that is sometimes breathtaking," but these rambling, obscure commentaries ("what's going to probably happen for the real Riemann Hypothesis is there's going to be another blob and there's going to be a function that turns the blob into itself") are not necessarily very helpful. Sabbagh can be a gifted expositor of mathematics when he sticks to more tractable topics, but when it comes to the Riemann Hypothesis, he offers readers veneration instead of understanding. B&w illustrations and graphs. (Apr.) Copyright 2003 Reed Business Information.

Library Journal

Thanks to a proof by Euclid, mathematicians have known for more than 2000 years that there is no limit to the population of prime numbers; they extend to infinity. However, work continues to be done on the distribution of the primes, and much of that work now centers on efforts to prove the Riemann hypothesis. Bernhard Riemann was a great 19th-century German mathematician who offered in an 1859 paper an admittedly unproven conjecture relating some zero values of a "zeta function" to the distribution of primes. The importance of this abstruse speculation for modern research is demonstrated in a recent online search of Mathematical Reviews for the term "Riemann hypothesis"; 1403 publications were found. Now there are three more books to add to the numerous studies. Derbyshire, a mathematician by training, a member of the Mathematical Association of America, and a novelist (Seeing Calvin Coolidge in a Dream), first takes readers through well-organized mathematical fundamentals in order to give them a good understanding of Riemann's discovery and its consequences. Interspersed with the hardcore math, other chapters profile Reimann the man and trace the history of mathematics in relation to his still-unproven hypothesis. Derbyshire shows how after 150 years, the world's greatest minds still haven't found a solution. Because this book does not sugarcoat complex ideas, readers lacking at least college-level math will be hard-pressed to understand some parts. Still, this volume is highly recommended for academic and larger public libraries as an excellent introduction for nonspecialists. Du Sautoy is the only professional research mathematician among these three authors, but he does not confront his readers with very many equations or other bits of mathematical apparatus. Instead, he offers nicely done verbal descriptions of the essence of the hypothesis and the efforts to prove it. Like Derbyshire, he intersperses items from math history and from the work and interactions of current researchers. Du Sautoy's book has much to offer for most academic and public libraries, especially to readers of very limited math background. Sabbagh (A Rum Affair) has written several books on a variety of topics, not all science-related. His latest emphasizes anecdotes from contemporary mathematicians who have studied Riemann's hypothesis. Indeed, he pays so much attention to a particularly idiosyncratic mathematician, ignored by the two other authors, that in his quest for human-interest material, he seems to lose sight of serious mathematical issues. Sabbagh's discussion of the actual mathematics is not so well organized, and much of it is relegated to a series of appendixes. His book is most useful in giving readers a feel for how research mathematicians live, work, and interrelate in the 21st century. Only libraries seeking comprehensive coverage of mathematics will need to get the Sabbagh work.-Jack W. Weigel, Ann Arbor, MI Copyright 2003 Reed Business Information.

Kirkus Reviews

British author Sabbagh (A Rum Affair, 2000, etc.) looks at a major unsolved problem in pure math and the men working to solve it. In 1859, the German mathematician Bernard Riemann stated his solution to the problem, which concerns the distribution of prime numbers in the natural number system. He could not provide a proof, but thought his answer "very probably true." Since then, the sharpest minds in math have wrestled with it-so far, without a proof. But its importance is such that experts believe that a definitive answer would instantly settle a hundred other unsolved problems that assume its truth as a starting point. The author uses the problem as an opportunity to profile some two dozen Riemann specialists. The result is a surprisingly warm portrait that focuses as much on these men's passion for mathematics and their reasons for becoming mathematicians as on the hypothesis itself. The central figure in this account is Purdue University's Louis de Branges, who may be on the verge of proving the Riemann hypothesis. But most of his peers doubt his claim, even though de Branges solved another difficult problem, the Beiderback conjecture, several years ago. Sabbagh provides a good look at the culture of world-class mathematicians: their rituals and their jokes, their politics and their shortcomings (many are only mediocre at day-to-day calculation). "Toolkits" appended to the text offer brief refreshers in the key mathematical concepts (equations, graphs, matrices, etc.) that the subjects here use. Even so, the problem remains unsolved unless the experts accept de Branges's proof, which is given in outline in an appendix. As the author admits, most readers will end up with no better ideaof the dimensions of Riemann's problem than before they started, but Sabbagh's picture of the mathematicians' world should amply compensate for that. A fine piece of scientific sociology.