October 2001

The Music of the Primes

As a teenager just starting to learn some real mathematics, one of the results that most amazed and intrigued me was Leonard Euler's result expressing the zeta function as an infinite product over the primes. Who could not look at this theorem and wonder what deep and profound mathematics lay beneath the equation Euler discovered?

Unfortunately, overshadowed by the complex version of the zeta function subsequently developed and used by Bernard Riemann, Euler's original real zeta function seems to have dropped out of sight in popular expositions of mathematics of late. With the hope of similarly inspiring another generation of future mathematicians, this month's column tries to rekindle interest in Euler's original and spectacular eighteenth century theorem.

To set the scene: Euler's theorem addresses one of the oldest questions of mathematics: What is the pattern of the primes numbers? Euclid devoted many pages of his mammoth work Elements to a treatment of prime numbers, including his famous result that the primes are infinite in number. Besides providing a proof of this fact completely different from Euclid's, Euler's zeta function theorem marked the beginning of the enormously important area of modern mathematics called analytic number theory, where methods of analysis are used to obtain results about whole numbers.

Because of the need to include quite a lot of mathematical formulas, I have prepared my account as a PDF file, which any modern web browser will open automatically using Acrobat Reader. Simply click on the link below to find out what Euler did and how he did it.

How Euler discovered the zeta function (PDF file)

Devlin's Angle is updated at the beginning of each month.
Mathematician Keith Devlin ( devlin@csli.stanford.edu) is the Executive Director of the Center for the Study of Language and Information at Stanford University and "The Math Guy" on NPR's Weekend Edition. His latest book is The Math Gene: How Mathematical Thinking Evolved and Why Numbers Are Like Gossip, published by Basic Books.