Analytic number theory

Exploring properties of the integers with complex analysis / From Wikipedia, the free encyclopedia

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In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers.[1] It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Dirichlet L-functions to give the first proof of Dirichlet's theorem on arithmetic progressions.[1][2] It is well known for its results on prime numbers (involving the Prime Number Theorem and Riemann zeta function) and additive number theory (such as the Goldbach conjecture and Waring's problem).

Riemann zeta function ζ(s) in the complex plane. The color of a point s encodes the value of ζ(s): colors close to black denote values close to zero, while hue encodes the value's argument.