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Number theory

Number theory was Dirichlet's main research interest, a field in which he found several deep results and in proving them introduced some fundamental tools, many of which were later named after him. In 1837, Dirichlet proved his theorem on arithmetic progressions using concepts from mathematical analysis to tackle an algebraic problem, thus creating the branch of analytic number theory. In proving the theorem, he introduced the Dirichlet characters and L-functions. In that article, he also noted the difference between the absolute and conditional convergence of series and its impact in what was later called the Riemann series theorem. In 1841, he generalized his arithmetic progressions theorem from integers to the ring of Gaussian integers Z {\displaystyle \mathbb {Z} }.

In a couple of papers in 1838 and 1839, he proved the first class number formula for quadratic forms (later refined by his student Kronecker). The formula, which Jacobi called a result "touching the utmost of human acumen", opened the way for similar results regarding more general number fields. Based on his research of the structure of the unit group of quadratic fields, he proved the Dirichlet unit theorem, a fundamental result in algebraic number theory.

He first used the pigeonhole principle, a basic counting argument, in the proof of a theorem in diophantine approximation, later named after him Dirichlet's approximation theorem. He published important contributions to Fermat's Last Theorem, for which he proved the cases n = 5 and n = 14, and to the biquadratic reciprocity law. The Dirichlet divisor problem, for which he found the first results by introducing the Dirichlet hyperbola method, is still an unsolved problem in number theory despite later contributions by other mathematicians.