Description of the book

The book has two aims:

• To explain the Hypothesis to readers who are not professional mathematicians.
• To give historical and biographical background matter — to humanize the Hypothesis.

So far as possible I have separated out these two aims.  Of the book's 22 chapters, the odd-numbered ones concentrate on the math, the even-numbered ones on the people and events.  It's not a completely watertight separation, but I hope that a reader who gives up on the math will be able to continue with the background material.

Bernhard Riemann was a German mathematician who lived from 1826 to 1866.  In 1859 he presented a paper to the Berlin Academy.  The title of the paper was: "On the Number of Prime Numbers less than a Given Quantity."

As the title suggests, the problem Riemann tackled in his paper was a straightforward matter of arithmetic.  How many prime numbers are there less than twenty?  Well, there are eight:  2, 3, 5, 7, 11, 13, 17 and 19.  How many are there less than a hundred?  Less than a trillion?  Is there a general rule for figuring out how many?  If so, what is it?

In the course of his inquiries, Riemann invented a particular mathematical object, the zeta function.  Three pages into the paper, he made a guess about that object.  Following the guess, he added:  "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."

There, a century and a half later, the matter still stands.  That guess became known as the Riemann Hypothesis.  It has, we now know, implications far and wide in mathematics and beyond — in nuclear physics, for example.  Nobody has been able to prove it true; nobody has been able to prove it false.  Whoever eventually proves either the one thing or the other will earn undying glory. 

As an object of mathematical interest, the Riemann Hypothesis was a "sleeper."  For thirty years following the 1859 paper, little attention was paid to it.  However, the techniques Riemann developed in that paper gave mathematicians a way to prove a lesser, but extremely important, theorem about the approximate distribution of prime numbers, the so-called Prime Number Theorem.  That proof was finally accomplished in 1896 by two mathematicians working independently, the Frenchman Jacques Hadamard and the Belgian Charles de la Vallée Poussin.

With the Prime Number Theorem out of the way, it quickly became clear that the Riemann Hypothesis was of central importance in pushing inquiries further.  It became an obsession of 20th-century mathematicians.  Towards the end of the century, with other long-standing problems like Fermat's Last Theorem finally solved, the Riemann Hypothesis became the "great white whale" of mathematics:  the deepest, most subtle and difficult, most long-standing of all mathematical problems — a Prime Obsession.

To give an idea of the scope of the historical and biographical side, here are the people whose photographs appear in the 8-page photo "well" at the center of the book.

Leonhard Euler (mathematician)
Peter the Great (Emperor of Russia)
Carl Friedrich Gauss (mathematician & physicist)
Carl Wilhelm Ferdinand, Duke of Brunswick (Gauss's patron)
Bernhard Riemann (young)
Bernhard Riemann (middle-aged)
Lejeune Dirichlet (mathematician)
Richard Dedekind (mathematician)
Charles de la Vallée Poussin (mathematician)
Jacques Hadamard (mathematician)
Pafnuty Lvovich Chebyshev (mathematician)
Atle Selberg (mathematician)
David Hilbert (mathematician)
Edmund Landau (mathematician)
G.H. Hardy (mathematician)
J.E. Littlewood (mathematician)
Jørgen Pedersen Gram (mathematician)
Carl Siegel (mathematician)
Alan Turing (mathematician)
Andrew Odlyzko (mathematician)
Emil Artin (mathematician)
André Weil (mathematician)
Pierre Deligne (mathematician)
Alain Connes (mathematician)
George Pólya (mathematician)
Freeman Dyson (physicist)
Hugh Montgomery (mathematician)
Sir Michael Berry (physicist)
Ernst Lindelöf (mathematician)
Harald Cramér (statistician)
John Derbyshire (couldn't resist the opportunity)