* Johann Peter Gustav Lejeune Dirichlet publishes a memoir giving the Dirichlet conditions, showing for which functions the convergence of the Fourier series holds ; introducing Dirichlet's test for the convergence of series ; the Dirichlet function as an example that not any function is integrable ; and, in the proof of the theorem for the Fourier series, the Dirichlet kernel and Dirichlet integral.

* Johann Peter Gustav Lejeune Dirichlet publishes Dirichlet's theorem on arithmetic progressions, using mathematical analysis concepts to tackle an algebraic problem and thus creating the branch of analytic number theory.

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