★★★★★ The Riemann Hypothesis explained

Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics

John Derbyshire

Bernhard Riemann was one of the greatest mathematicians of all time. Indeed, he is a strong candidate for GOAT. Mathematics, John Derbyshire tells us, is traditionally divided into four subdisciplines: arithmetic, geometry, algebra, and analysis. Riemann is most famous for his work in analysis. Analysis is the mathematics of the continuum. In addition Riemann made contributions to geometry -- he was one of the discoverers of non-Euclidean geometry, which would eventually become the basis of Einstein's Theory of General Relativity. And, most surprisingly, Riemann discovered a deep connection between analysis and arithmetic, number theory, the prime numbers. He set this out in one 1859 paper entitled "Über die Anzahl der Primzahlen unterhalb einer gegebenen Größe" -- "On the number of prime numbers below a given size". Within that paper he made a guess, which he couldn't prove, and wrote

One would, of course, like to have a rigorous proof of this, but I have put aside the search for such a proof after some fleeting vain attempts because it is not necessary for the immediate objective of my investigation.

Over 160 years later that guess, now known as the Riemann Hypothesis, remains unproved. Derbyshire writes, accurately I think, "The Riemann Hypothesis is now the great white whale of mathematical research."

What is the Riemann Hypothesis? It is a very technical conjecture about a mathematical object known as the Riemann zeta function. I cannot explain it briefly. At least half of Prime Obsession is devoted to that purpose. You're not going to get a comprehensible shorter answer than by reading this book. I thoroughly enjoyed it.

Make no mistake -- this is a book about math. If you don't like math, you should probably not attempt it. However, it is also a book about mathematicians. The odd-numbered chapters cover the mathematics, while the even-numbered chapters cover history and biography. You could presumably read the even chapters alone to learn something about the Riemann Hypothesis while avoiding all the math. I didn't try that, so I can't tell you how well it would work.

Derbyshire is careful to describe himself as a journalist, not a mathematician. However, he obviously knows a great deal about mathematics and is good at explaining it. I read The Music of the Primes at the same time as my first reading of Prime Obsession. The Music of the Primes is an attempt by Marcus du Sautoy, a card-carrying mathematician, to do what Derbyshire has done in Prime Obsession. I was surprised to find that Prime Obsession is much the better of the two books. It is not only that Derbyshire explains better -- his explanations are also, surprisingly, more mathematically rigorous than du Sautoy's.

I was particularly impressed with Derbyshire's handling of the theorem he grandly calls "The Golden Key", also known as Euler's product formula. He presents a complete and very clear proof. He explains how he found this proof as follows

When jotting down the ideas that make up this book, I first looked through some of the math texts on my shelves to find a proof of the Golden Key suitable for non-specialist readers. I settled on one that seemed to me acceptable and incorporated it. At a later stage of the book's development, I thought I had better carry out authorial due diligence, so I went to a research library (in this case the excellent new Science, Industry and Business branch of the New York Public Library in midtown Manhattan) and pulled out the original paper from Euler's collected works. His proof of the Golden Key covers ten lines and is far easier and more elegant than the one I had selected from my textbooks. I thereupon threw out my first choice of proof and replaced it with Euler's. The proof in part III of this chapter is essentially Euler's. It's a professorial cliché, I know, but it's true nonetheless: you can't beat going to the original sources.

Derbyshire's exposition of Euler's proof covers far more than ten lines -- Euler was writing for mathematicians and could abbreviate, knowing his readers would fill in the gaps. Derbyshire makes no such assumption and his proof is a thing of beauty.

Mathematicians are concerned, more than any other profession I know (including the arts) with the pursuit of beauty. Nonmathematicians are often surprised to hear this -- they don't perceive beauty in mathematics. Derbyshire has done an outstanding job of presenting mathematics as the beautiful thing it is.

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