analytic number theory

E530316

branch of mathematics subfield of number theory

Analytic number theory is a branch of mathematics that uses tools from mathematical analysis to study the distribution and properties of integers, especially prime numbers.

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Statements (58)

Predicate	Object
instanceOf	branch of mathematics ⓘ subfield of number theory ⓘ
appliesTo	Diophantine equations NERFINISHED ⓘ arithmetic geometry ⓘ problems about prime distribution ⓘ
centralConjecture	Riemann hypothesis NERFINISHED ⓘ
centralObject	Riemann zeta function NERFINISHED ⓘ
centralTheorem	Bombieri–Vinogradov theorem NERFINISHED ⓘ Chebotarev density theorem NERFINISHED ⓘ Dirichlet's theorem on arithmetic progressions NERFINISHED ⓘ Pólya–Vinogradov inequality NERFINISHED ⓘ Siegel–Walfisz theorem NERFINISHED ⓘ Weyl's equidistribution theorem NERFINISHED ⓘ prime number theorem NERFINISHED ⓘ zero-free regions for L-functions ⓘ
focusesOn	additive properties of integers ⓘ distribution of integers in arithmetic progressions ⓘ distribution of prime numbers ⓘ multiplicative properties of integers ⓘ
hasSubarea	additive number theory (analytic methods) ⓘ analytic theory of L-functions ⓘ multiplicative number theory ⓘ sieve theory ⓘ
historicalFigure	Atle Selberg NERFINISHED ⓘ Bernhard Riemann NERFINISHED ⓘ Charles-Jean de la Vallée Poussin NERFINISHED ⓘ G. H. Hardy NERFINISHED ⓘ Harald Cramér NERFINISHED ⓘ Ivan Vinogradov NERFINISHED ⓘ Jacques Hadamard NERFINISHED ⓘ John Edensor Littlewood NERFINISHED ⓘ
relatedField	additive combinatorics ⓘ algebraic number theory ⓘ probabilistic number theory ⓘ
studies	Goldbach-type problems ⓘ Waring-type problems ⓘ additive problems in number theory ⓘ automorphic forms ⓘ distribution of primes in arithmetic progressions ⓘ distribution of primes in short intervals ⓘ distribution of values of arithmetic functions ⓘ integers ⓘ modular forms ⓘ prime gaps ⓘ prime numbers ⓘ zeros of L-functions ⓘ zeros of the Riemann zeta function ⓘ
usesTool	Dirichlet series NERFINISHED ⓘ Fourier analysis NERFINISHED ⓘ L-functions ⓘ Tauberian theorems NERFINISHED ⓘ complex analysis ⓘ generating functions ⓘ harmonic analysis ⓘ mathematical analysis ⓘ probabilistic methods ⓘ sieve methods ⓘ zeta functions ⓘ

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Fermat's theorem on sums of two squares → usedIn → analytic number theory ⓘ