# Dirichlet, Peter Gustav Lejeune

### theory series conditions gauss’s

[deereesh lay ] (1805–59) German mathematician: contributed to analysis, partial differential equations in physics and number theory.

Dirichlet studied at Göttingen under and also spent time in Paris, where he gained an interest in series from their originator. He moved to a post at Breslau but at 23 became a professor at Berlin, remaining for 27 years. He was shy and modest but an excellent teacher; he was a close friend of Jacobi and spent 18 months in Italy with him when Jacobi was driven there by ill-health. On Gauss’s death in 1855 Dirichlet accepted his prestigious chair at Göttingen but died of a heart attack only 3 years later.

Dirichlet carried on Gauss’s great work on number theory, publishing on Diophantine equations of the form *x* 5 + *y* 5 = *kz* 5 , and developing a general algebraic number theory. Dirichlet’s theorem (1837) states that any arithmetic series *a* , *a* + *b* , *a* + 2 *b* , *a* + 3 *b* , where *a* and *b* have no common divisors other than 1, must include an infinite series of primes. His book *Lectures on Number Theory* (1863) is a work of similar stature to Gauss’s earlier *Disquisitiones* and founded modern algebraic number theory.

Dirichlet also made advances in applied mathematics. In 1829 he stated the conditions sufficient for a Fourier series to converge (those conditions necessary for it to converge are still undiscovered). He worked on multiple integrals and the boundary-value problem (or Dirichlet problem), which is the effect of the conditions at the boundary on the solution of a heat flow or electrostatic equation.

It is not only Dirichlet’s many specific contributions that give him greatness, but also his approach to formulating and analysing problems for which he founded modern techniques.

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