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 Results6 entries

Chebyshev Function

Analytic Number Theory

Dirichlet Eta Function

Analytic Number Theory

Logarithmic Integral Function

Analytic Number Theory

Perron's Formula

Analytic Number Theory

Riemann Zeta Function

Analytic Number Theory

Z Function

Analytic Number Theory
Connections

Associated With

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

Includes

A. I. Vinogradov, Mathematicians
Source Askold Vinogradov (Wikipedia)
Abel's Summation Formula, Mathematical Objects
Source Abel's Summation Formula (Wikipedia)
Anatoly Karatsuba, Mathematicians
Source Anatoly Karatsuba (Wikipedia)
Atle Selberg, Mathematicians
Source Atle Selberg, Biography (MacTutor History of Mathematics)
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
Chebyshev Function, Mathematical Objects
Source Chebyshev Function (Wikipedia)
Chen Jingrun, Mathematicians
Source Chen Jingrun (Wikipedia)
Dirichlet Eta Function, Mathematical Objects
Source Dirichlet Eta Function (Wikipedia)
Edmund Landau, Mathematicians
Source Edmund Landau, Biography (MacTutor History of Mathematics)
Edward Charles Titchmarsh, Mathematicians
Source Wikipedia: Edward Charles Titchmarsh
Enrico Bombieri, Mathematicians
Source MacTutor biography of Enrico Bombieri, St Andrews
Friedlander-Iwaniec Theorem, Theorems
Source Friedlander-Iwaniec theorem (Wikipedia)
G. H. Hardy, Mathematicians
Source Godfrey Harold Hardy (MacTutor)
Hans Rademacher, Mathematicians
Source Hans Rademacher, Biography (MacTutor History of Mathematics)
Hans-Egon Richert, Mathematicians
Source Wikipedia: Hans-Egon Richert
Harold Davenport, Mathematicians
Source Harold Davenport (MacTutor)
Heini Halberstam, Mathematicians
Source Heini Halberstam (Wikipedia)
Helmut Maier, Mathematicians
Source Wikipedia: Helmut Maier
Henryk Iwaniec, Mathematicians
Source Henryk Iwaniec (Wikipedia)
Hugh Montgomery, Mathematicians
Source University of Michigan Board of Regents: Hugh L. Montgomery retirement memoir (University of Michigan Regents)
Ivan Vinogradov, Mathematicians
Source Ivan Vinogradov (Wikipedia)
Jacques Hadamard, Mathematicians
John Friedlander, Mathematicians
Source John Friedlander (Wikipedia)
Jurkat-Richert Theorem, Theorems
Source Jurkat-Richert theorem (Wikipedia)
Linnik's Theorem, Theorems
Source Linnik's theorem (Wikipedia)
Logarithmic Integral Function, Mathematical Objects
Source Logarithmic Integral Function (Wikipedia)
Nikolai Korobov, Mathematicians
Source Nikolay Korobov (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 of the proof of the asymptotic law of prime numbers
Riemann Hypothesis, Conjectures
Riemann Zeta Function, Mathematical Objects
Source Riemann Zeta Function (Wikipedia)
Siegel-Walfisz Theorem, Theorems
Source Siegel-Walfisz theorem, Wikipedia
Theodor Estermann, Mathematicians
Source Theodor Estermann biography (MacTutor)
Vinogradov's Mean-Value Theorem, Theorems
Source Vinogradov's mean value theorem (Wikipedia)
Z Function, Mathematical Objects
Source Z Function (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.
  • In Group: Algebraic and Analytic Number Theory, lead paragraph
    In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers.
View the Source
Prime Number Theorem (Wikipedia)
Wikimedia FoundationIncludes: Prime Number Theorem, History of the proof of the asymptotic law of prime numbers
Quote, Includes: Prime Number Theorem, History of the proof of the asymptotic law of prime numbers
In particular, it is in this paper that the idea to apply methods of complex analysis to the study of the real function π(x) originates.
View the Source
Riemann Zeta Function (Wikipedia)
Wikimedia FoundationIncludes: Riemann Zeta FunctionView the Source
Logarithmic Integral Function (Wikipedia)
Includes: Logarithmic Integral Function, Lead
Quote, Includes: Logarithmic Integral Function, Lead
It is relevant in problems of physics and has number theoretic significance
View the Source
University of Michigan Board of Regents: Hugh L. Montgomery retirement memoir (University of Michigan Regents)
University of Michigan Board of RegentsIncludes: Hugh Montgomery, Memoir
Quote, Includes: Hugh Montgomery, Memoir
He specialized in analytic number theory
View the Source
Harold Davenport (MacTutor)
Includes: Harold Davenport, Biography, analytic theory
Quote, Includes: Harold Davenport, Biography, analytic theory
the analytic theory of numbers
View the Source
Godfrey Harold Hardy (MacTutor)
Includes: G. H. Hardy, Summary
Quote, Includes: G. H. Hardy, Summary
Hardy 's interests covered many topics of pure mathematics:- Diophantine analysis, summation of divergent series, Fourier series, the Riemann zeta function and the distribution of primes.
View the Source
Chen Jingrun (Wikipedia)
Includes: Chen Jingrun, Life and career, research
Quote, Includes: Chen Jingrun, Life and career, research
led to progress in analytic number theory
View the Source
Henryk Iwaniec (Wikipedia)
Includes: Henryk Iwaniec, Lead
Quote, Includes: Henryk Iwaniec, Lead
He has made important contributions to analytic and algebraic number theory as well as harmonic analysis.
View the Source
Theodor Estermann biography (MacTutor)
Includes: Theodor Estermann, Summary
Quote, Includes: Theodor Estermann, Summary
Theodor Estermann was a German mathematician who worked in the field of analytic number theory.
View the Source
Heini Halberstam (Wikipedia)
Includes: Heini Halberstam, Lead, field
Quote, Includes: Heini Halberstam, Lead, field
working in the field of analytic number theory
View the Source
John Friedlander (Wikipedia)
Includes: John Friedlander, Lead
Quote, Includes: John Friedlander, Lead
is a Canadian mathematician specializing in analytic number theory
View the Source
Askold Vinogradov (Wikipedia)
Includes: A. I. Vinogradov, Lead, field of work
Quote, Includes: A. I. Vinogradov, Lead, field of work
who worked in analytic number theory
View the Source
Anatoly Karatsuba (Wikipedia)
Includes: Anatoly Karatsuba, Lead, field
Quote, Includes: Anatoly Karatsuba, Lead, field
was a Russian mathematician working in the field of analytic number theory, p-adic numbers and Dirichlet series
View the Source
Nikolay Korobov (Wikipedia)
Includes: Nikolai Korobov, Lead, best known work
Quote, Includes: Nikolai Korobov, Lead, best known work
He is best known for his work in analytic number theory
View 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
Wikipedia: Edward Charles Titchmarsh
Includes: Edward Charles Titchmarsh, Lead
Quote, Includes: Edward Charles Titchmarsh, Lead
Titchmarsh was known for work in analytic number theory
View the Source
Wikipedia: Hans-Egon Richert
Includes: Hans-Egon Richert, Lead
Quote, Includes: Hans-Egon Richert, Lead
Hans-Egon Richert (June 2, 1924, November 25, 1993) was a German mathematician who worked primarily in analytic number theory.
View the Source
Wikipedia: Helmut Maier
Includes: Helmut Maier, Lead
Quote, Includes: Helmut Maier, Lead
He is known for his contributions in analytic number theory and mathematical analysis and particularly for the so-called Maier's matrix method as well as Maier's theorem for primes in short intervals.
View the Source
Vinogradov's mean value theorem (Wikipedia)
Includes: Vinogradov's Mean-Value Theorem, Lead paragraph
Quote, Includes: Vinogradov's Mean-Value Theorem, Lead paragraph
It is an important inequality in analytic number theory, named for I.
View the Source
Chebyshev Function (Wikipedia)
Includes: Chebyshev Function, Lead section
Quote, Includes: Chebyshev Function, Lead section
the natural logarithm, with the sum extending over all prime numbers p that are less than or equal to x. The second
View the Source
Jurkat-Richert theorem (Wikipedia)
Includes: Jurkat-Richert Theorem, Lead section
Quote, Includes: Jurkat-Richert Theorem, Lead section
The Jurkat-Richert theorem is a mathematical theorem in sieve theory. It is a key ingredient in proofs of Chen's theorem
View the Source
Z Function (Wikipedia)
Includes: Z Function, Lead section
Quote, Includes: Z Function, Lead section
the Z function is a function used for studying the Riemann zeta function along the critical line where the argument is
View the Source
MacTutor biography of Enrico Bombieri, St Andrews
Includes: Enrico Bombieri, Summary
Quote, Includes: Enrico Bombieri, Summary
Enrico Bombieri is an Italian Fields medal winner, known for his work in analytic number theory, algebraic geometry, theory of several complex variables, minimal surfaces and the theory of finite groups.
View the Source
Atle Selberg, Biography (MacTutor History of Mathematics)
Includes: Atle Selberg, Summary
Quote, Includes: Atle Selberg, Summary
Atle Selberg was a Norwegian mathematician known for his work in analytic number theory and on discrete groups and automorphic forms.
View the Source
Edmund Landau, Biography (MacTutor History of Mathematics)
Includes: Edmund Landau, Summary
Quote, Includes: Edmund Landau, Summary
Landau gave the first systematic presentation of analytic number theory and wrote important works on the theory of analytic functions of a single variable.
View the Source
Hans Rademacher, Biography (MacTutor History of Mathematics)
Includes: Hans Rademacher, Summary
Quote, Includes: Hans Rademacher, Summary
Hans Rademacher was a German-born mathematician who worked in analytic number theory, mathematical genetics, the theory of functions of a real variable and quantum theory.
View the Source
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