Analytic number theory is the branch of number theory that uses methods from mathematical analysis, above all the study of the prime numbers and how they are distributed among the integers. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
Central QuestionHow the prime numbers are distributed among the integers, a question the prime number theorem answers asymptotically by showing that x divided by the natural logarithm of x closely approximates the count of primes up to x as x grows without bound. 1 Key DebateWhether the Riemann Hypothesis, Bernhard Riemann's conjecture about the location of the zeros of the zeta function, is true. Jacques Hadamard and Charles-Jean de la Vallee Poussin independently proved the weaker form of Riemann's conjectures needed to establish the prime number theorem in 1896, but the Hypothesis itself, a much sharper claim about how closely the primes track their expected distribution, remains one of the most famous unsolved problems in mathematics. 1 Classification
Pure or Applied Analytic Number Theory
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Source Abel's Summation Formula (Wikipedia)
Source Bombieri-Vinogradov Theorem (Wikipedia)
Source Brun-Titchmarsh theorem (Wikipedia)
Source Carl Ludwig Siegel (Wikipedia)
Source Dirichlet Eta Function (Wikipedia)
Source Friedlander-Iwaniec theorem (Wikipedia)
Source Ivan Vinogradov (Wikipedia)
Source Linnik's theorem (Wikipedia)
Source Perron's Formula (Wikipedia)
Source Peter Gustav Lejeune Dirichlet (Wikipedia)
Additional Source Prime Number Theorem (Wikipedia)History section
Source Riemann Zeta Function (Wikipedia)
Source Siegel-Walfisz theorem, Wikipedia
Sources
1. Analytic Number Theory (Wikipedia)
WikipediaOpening paragraph
In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers.
Precursors
The prime number theorem then states that x / ln(x) is a good approximation to pi(x), in the sense that the limit of the quotient of the two functions pi(x) and x / ln(x) as x approaches infinity is 1
Riemann
He made a series of conjectures about properties of the zeta function, one of which is the well-known Riemann hypothesis.
View the Source Prime Number Theorem (Wikipedia)
Wikimedia FoundationIncludes: Prime Number Theorem, History sectionQuote, Includes: Prime Number Theorem, History section
it is in this paper that the idea to apply methods of complex analysis to the study of the real function pi(x) originates
Riemann Zeta Function (Wikipedia)
Abel's Summation Formula (Wikipedia)
Includes: Abel's Summation Formula, Lead sentenceQuote, Includes: Abel's Summation Formula, Lead sentence
uced by Niels Henrik Abel, is intensively used in analytic number theory and the study of special functions to compute series.
Perron's Formula (Wikipedia)
Includes: Perron's Formula, Lead sentenceQuote, Includes: Perron's Formula, Lead sentence
In mathematics, and more particularly in analytic number theory, Perron's formula is a formula discovered by Oskar Perron to calcu
Dirichlet Eta Function (Wikipedia)
Includes: Dirichlet Eta Function, Lead sentenceQuote, Includes: Dirichlet Eta Function, Lead sentence
In mathematics, in the area of analytic number theory, the Dirichlet eta function is defined by the following Dirichlet series, wh
Siegel-Walfisz theorem, Wikipedia
Bombieri-Vinogradov Theorem (Wikipedia)
Wikimedia FoundationIncludes: Bombieri-Vinogradov Theorem, Lead sentenceQuote, Includes: Bombieri-Vinogradov Theorem, Lead sentence
y called Bombieri's theorem) is a major result of analytic number theory, obtained in the mid-1960s, concerning the distribution o
Brun-Titchmarsh theorem (Wikipedia)
Linnik's theorem (Wikipedia)
Includes: Linnik's Theorem, Lead sentenceQuote, Includes: Linnik's Theorem, Lead sentence
Linnik's theorem in analytic number theory answers a natural question after Dirichlet's theorem on arithmetic progressions.
Friedlander-Iwaniec theorem (Wikipedia)
Carl Ludwig Siegel (Wikipedia)
Includes: Carl Ludwig Siegel, Lead paragraph [in-branch]Quote, Includes: Carl Ludwig Siegel, Lead paragraph [in-branch]
analytic number theory
Peter Gustav Lejeune Dirichlet (Wikipedia)
Includes: Peter Gustav Lejeune Dirichlet, Lead paragraph [in-branch 2]Quote, Includes: Peter Gustav Lejeune Dirichlet, Lead paragraph [in-branch 2]
created analytic number theory
Ivan Vinogradov (Wikipedia)
Includes: Ivan Vinogradov, Lead paragraph [in-branch]Quote, Includes: Ivan Vinogradov, Lead paragraph [in-branch]
analytic number theory
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