| | Devlin's Angle |
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.