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An analogue of Dirichlet's theorem holds in the framework of dynamical systems (T. Sunada and A. Katsuda, 1990). Shiu showed that any arithmetic progression satisfying the hypothesis of Dirichlet's theorem will in fact contain arbitrarily long runs of consecutive prime numbers. [7]
The Vorlesungen contains two key results in number theory which were first proved by Dirichlet. The first of these is the class number formulae for binary quadratic forms. The second is a proof that arithmetic progressions contains an infinite number of primes (known as Dirichlet's theorem); this proof introduces Dirichlet L-series. These ...
In 1837 he published Dirichlet's theorem on arithmetic progressions, using mathematical analysis concepts to tackle an algebraic problem and thus creating the branch of analytic number theory. In proving the theorem, he introduced the Dirichlet characters and L-functions.
Dirichlet's theorem may refer to any of several mathematical theorems due to Peter Gustav Lejeune Dirichlet. Dirichlet's theorem on arithmetic progressions; Dirichlet's approximation theorem; Dirichlet's unit theorem; Dirichlet conditions; Dirichlet boundary condition; Dirichlet's principle; Pigeonhole principle, sometimes also called Dirichlet ...
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.
For example, in proving Dirichlet's theorem on arithmetic progressions, it is easy to show that the set of primes in an arithmetic progression a + nb (for a, b coprime) has Dirichlet density 1/φ(b), which is enough to show that there are an infinite number of such primes, but harder to show that this is the natural density.
Dirichlet character; Dirichlet L-series. Siegel zero; Dirichlet's theorem on arithmetic progressions. Linnik's theorem; Elliott–Halberstam conjecture; Functional equation (L-function) Chebotarev's density theorem; Local zeta function. Weil conjectures; Modular form. modular group; Congruence subgroup; Hecke operator; Cusp form; Eisenstein ...
See also Dirichlet's theorem on arithmetic progressions. As of 2020, the longest known arithmetic progression of primes has length 27: [4] 224584605939537911 + 81292139·23#·n, for n = 0 to 26. (23# = 223092870) As of 2011, the longest known arithmetic progression of consecutive primes has length 10. It was found in 1998.