Dirichlet

Peter Gustav Lejeune Dirichlet (1805-1859)

Peter Gustav Lejeune Dirichlet was born in Düren, then in the French Empire, but now in western Germany, on 13 February 1805 and was educated at the University of Göttingen, where Carl Friedrich Gauss was one of his mentors. He was fluent in both French and German and as such was often involved in communicating ideas between French and Geman mathematicians.

He made major contributions in the fields of number theory, analysis and mechanics, and taught in the Universities of Breslau (1827) and Berlin (1828-1855) before succeeding Gauss at the University of Göttingen.

It was Dirichlet who proposed (in 1837) the Theorem in his name which states the exisence of an infinite number of primes in any arithmetic series a+b, 2a+b, 3a+b, ..., na+b, in which neither of a nor b are divisible by the other. For example, 5, 11, 17, 23 and 29 are among the primes of the form 6n+5.

Independently, he and Legendre independently proved Fermat's Last Theorem for the case n=5, reportedly using an idea suggested by Sophie Germain. Actually, Dirichlet's proof was published in 1825 and reportedly had an error which was corrected by Legendre.

He developed the theory of units in algebraic number theory and made major contributions to the theory of ideals.

In 1837 he introduced the modern concept of a function with notation y=f(x) in which y is uniquely determined by the value of x. This work was inspired by significant contributions he had made to the understanding on Fourier Series, particularly with respect to conditions of convergence.

In mechanics he investigated the solutions of boundary problems with partial differential equations in the interior of a region. If the unknown is prescribed on the boundary of the region the problem is known as a Dirichlet problem. The minimisation principle with respect to certain classes of boundary value problems is oftern known as Dirichlet's minimisation principle.

It was Dirichlet who formulated the Pigeonhole Principle, often known as Dirichlet's Principle, which states

If there are p pigeons placed in h holes and p>h then there must be at least one pigeonhole containing at least 2 pigeons.

It was he who underlined the use of its systematical application as a powerful tool for creating mathematical proofs. Dirichlet's Principle enables short and elegant solutions of problems, previously attacked by complicated constructive methods.

Dirichlet died on 5 May 1859, in Göttingen.

Written by Peter Taylor, January 2001.

A number of historical references were used in composing this short biography. They included

• Men of Mathematics, ET Bell, Simon and Schuster, New York, 1937, 1965.
• An Introduction to the History of Mathematics, Howard Eves, 4th ed,, Holt, Rinehart and Winston, New York, 1976.

Dirichlet is the subject of the Australian Mathematics Trust T Shirt in 2001.

This is available for sale through the AMT.