Mathematics Atlas

How Proof Is Made
Sign In
Text size
100%
Theme

Switch Atlas

Branches of Mathematic

Analytic Number Theory

Number Theory

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 Question
How 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 Debate
Whether 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
Pure Mathematics 1
Analytic Number Theory
Filter Results1 entry

Abel's Summation Formula

Analytic Number Theory
Connections

Associated With

Algebraic Number Theory, Branches of Mathematics
Number Theory, Branches of Mathematics

Includes

Abel's Summation Formula, Mathematical Objects
Source Abel's Summation Formula (Wikipedia)
Bernhard Riemann, Mathematicians
Bombieri-Vinogradov Theorem, Theorems
Source Bombieri-Vinogradov Theorem (Wikipedia)
Brun-Titchmarsh Theorem, Theorems
Source Brun-Titchmarsh theorem (Wikipedia)
Carl Ludwig Siegel, Mathematicians
Source Carl Ludwig Siegel (Wikipedia)
Charles-Jean de la Vallee Poussin, Mathematicians
Dirichlet Eta Function, Mathematical Objects
Source Dirichlet Eta Function (Wikipedia)
Friedlander-Iwaniec Theorem, Theorems
Source Friedlander-Iwaniec theorem (Wikipedia)
Ivan Vinogradov, Mathematicians
Source Ivan Vinogradov (Wikipedia)
Jacques Hadamard, Mathematicians
Linnik's Theorem, Theorems
Source Linnik's theorem (Wikipedia)
Perron's Formula, Mathematical Objects
Source Perron's Formula (Wikipedia)
Peter Gustav Lejeune Dirichlet, Mathematicians
Source Peter Gustav Lejeune Dirichlet (Wikipedia)
Prime Number Theorem, Theorems
Additional Source Prime Number Theorem (Wikipedia)History section
Riemann Hypothesis, Conjectures
Riemann Zeta Function, Mathematical Objects
Source Riemann Zeta Function (Wikipedia)
Siegel-Walfisz Theorem, Theorems
Source Siegel-Walfisz theorem, Wikipedia
Sources
1. Analytic Number Theory (Wikipedia)
Wikipedia
  • Opening 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 section
Quote, 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
View the Source
Riemann Zeta Function (Wikipedia)
Wikimedia FoundationIncludes: Riemann Zeta FunctionView the Source
Abel's Summation Formula (Wikipedia)
Includes: Abel's Summation Formula, Lead sentence
Quote, 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.
View the Source
Perron's Formula (Wikipedia)
Includes: Perron's Formula, Lead sentence
Quote, 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
View the Source
Dirichlet Eta Function (Wikipedia)
Includes: Dirichlet Eta Function, Lead sentence
Quote, 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
View the Source
Siegel-Walfisz theorem, Wikipedia
Includes: Siegel-Walfisz Theorem, Lead sentenceView the Source
Bombieri-Vinogradov Theorem (Wikipedia)
Wikimedia FoundationIncludes: Bombieri-Vinogradov Theorem, Lead sentence
Quote, 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
View the Source
Brun-Titchmarsh theorem (Wikipedia)
Includes: Brun-Titchmarsh Theorem, Lead sentenceView the Source
Linnik's theorem (Wikipedia)
Includes: Linnik's Theorem, Lead sentence
Quote, Includes: Linnik's Theorem, Lead sentence
Linnik's theorem in analytic number theory answers a natural question after Dirichlet's theorem on arithmetic progressions.
View the Source
Friedlander-Iwaniec theorem (Wikipedia)
Includes: Friedlander-Iwaniec Theorem, Lead sentenceView the Source
Carl Ludwig Siegel (Wikipedia)
Includes: Carl Ludwig Siegel, Lead paragraph [in-branch]
Quote, Includes: Carl Ludwig Siegel, Lead paragraph [in-branch]
analytic number theory
View the Source
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
View the Source
Ivan Vinogradov (Wikipedia)
Includes: Ivan Vinogradov, Lead paragraph [in-branch]
Quote, Includes: Ivan Vinogradov, Lead paragraph [in-branch]
analytic number theory
View the Source
Comments (0)
No comments yet. Be the first to share a thought.
Reader Challenges (0)
No disputes yet. Spotted an error or a better source? Open the first one.