Born in Düren (now in Germany), Dirichlet studied mathematics at Göttingen where he was a pupil of Karl Gauss and Karl Jacobi. He also studied briefly in Paris where he met Joseph Fourier, who stimulated his interest in trigonometric series. In 1826 he returned to Germany and taught at Breslau and later at the Military Academy in Berlin. He then moved to the University of Berlin, which he only left 27 years later when he returned to Göttingen to fill the chair left vacant by Gauss's death.

Dirichlet's work in number theory was very much inspired by Gauss's great work in that field, and Dirichlet's own book, the Vorlesungen über Zahlentheorie (1863; Lectures on Number Theory), is of comparable historical importance to Gauss's Disquisitiones. He made many very significant discoveries in the field and his work on a problem connected with primes led him to make the fundamentally important innovation of using analytical techniques to obtain results in number theory.

His stay in Paris had stimulated Dirichlet's interest in Fourier series and in 1829 he was able to solve the outstanding problem of stating the conditions sufficient for a Fourier series to converge. (The other problem of giving necessary conditions is still unsolved). Fourier also gave the young Dirichlet an interest in mathematical physics, which led him to important work on multiple integrals and the boundary-value problem, now known as the Dirichlet problem, concerning the formulation and solution of those partial differential equations occurring in the study of heat flow and electrostatics. These are of great importance in many other areas of physics. The growth of a more rigorous understanding of analysis owes to Dirichlet what is essentially the modern definition of the concept of a function.