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

by John Derbyshire

The plan of the book is very simple. The odd-numbered chapters (I was going to make it the prime-numbered, but there is such a thing as being too cute) contain mathematical exposition, leading the reader, gently I hope, to

an understanding of the Riemann Hypothesis and its importance. The even-numbered chapters offer historical and biographical background matter.

The traditional division of mathematics into subdisciplines is as follows. Arithmetic—The study of whole numbers and fractions. Sample theorem: If you subtract an odd number from an even number you get an odd number. Geometry—The study of figures in space—points, lines, curves, and three-dimensional objects. Sample theorem: The angles of a triangle on a flat surface add up to 180 degrees. Algebra—The use of abstract symbols to represent mathematical objects (numbers, lines, matrices, transformations), and the study of the rules for combining those symbols. Sample theorem: For any two numbers x and y, (x + y) × (x – y) = x2 – y2. Analysis—The study of limits. Sample theorem: The harmonic series is divergent (that is, it increases without limit).

There is a story about the English mathematician G.H. Hardy, whom we shall meet later. In the middle of delivering a lecture, Hardy arrived at a point in his argument where he said, “It is now obvious that….” Here he stopped, fell silent, and stood motionless with furrowed brow for a few seconds. Then he walked out of the lecture hall. Twenty minutes later he returned, smiling, and began, “Yes, it is obvious that….”

V. What, you may be wondering, is so all-fired special about Expression 7-3, that I have given it such a grandiloquent name? The answer to that won't become entirely clear until a later chapter, when I actually turn the Golden Key. At this point, the main thing to be impressed by—mathematicians, at any rate, find it extremely impressive—is the fact that on the left-hand side of Expression 7-3 we have an infinite sum running through all the positive whole numbers 1, 2, 3, 4, 5, 6, …, while on the right-hand side we have an infinite product running through all the prime numbers 2, 3, 5, 7, 11, 13, ….

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.

[T]he answer appears to us before the question….

Every morning I would sit down before a blank sheet of paper. Throughout the day, with a brief interval for lunch, I would stare at the blank sheet. Often when evening came it was still empty…. [T]he two summers of 1903 and 1904 remain in my mind as a period of complete intellectual deadlock…. [I]t seemed quite likely that the whole of the rest of my life might be consumed in looking at that blank sheet of paper.