In analysis, he was one of the first mathematicians who advocated strict methods. 1837, he put forward the modern viewpoint that function is the corresponding relationship between x and y.
In number theory, he is the disseminator and disseminator of Gauss thought. 1833, Dirichlet wrote "Lecture Notes on Number Theory", which clearly explained Gauss's epoch-making work "Arithmetic Research" and made his thoughts widely spread. 1837, he constructed Dirichlet series. From 1838 to 1839, he got the formula to determine the class number of quadratic forms. In 1846, the pigeon hole principle is used. The structure of Abel group of unit number in algebraic number field is expounded.
In mathematical physics, he has done important work on topics such as gravity produced by ellipsoid, motion of ball in incompressible fluid and general stability derived from the stability of solar system. 1850 published an article on potential theory and discussed the famous first boundary value problem, which is now called Dirichlet problem.