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 Connections
Associated With
Includes
Source Abel's Summation Formula (Wikipedia)
Source Atle Selberg, Biography (MacTutor History of Mathematics)
Source Bombieri-Vinogradov Theorem (Wikipedia)
Source Brun-Titchmarsh theorem (Wikipedia)
Source Carl Ludwig Siegel (Wikipedia)
Source Chebyshev Function (Wikipedia)
Source Dirichlet Eta Function (Wikipedia)
Source Edmund Landau, Biography (MacTutor History of Mathematics)
Source Wikipedia: Edward Charles Titchmarsh
Source MacTutor biography of Enrico Bombieri, St Andrews
Source Friedlander-Iwaniec theorem (Wikipedia)
Source Hans Rademacher, Biography (MacTutor History of Mathematics)
Source Wikipedia: Hans-Egon Richert
Source Wikipedia: Helmut Maier
Source University of Michigan Board of Regents: Hugh L. Montgomery retirement memoir (University of Michigan Regents)
Source Ivan Vinogradov (Wikipedia)
Source Jurkat-Richert theorem (Wikipedia)
Source Linnik's theorem (Wikipedia)
Source Logarithmic Integral Function (Wikipedia)
Source Perron's Formula (Wikipedia)
Source Peter Gustav Lejeune Dirichlet (Wikipedia)
Additional Source Prime Number Theorem (Wikipedia)History of the proof of the asymptotic law of prime numbers
Source Riemann Zeta Function (Wikipedia)
Source Siegel-Walfisz theorem, Wikipedia
Source Vinogradov's mean value theorem (Wikipedia)
Source Z Function (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 of the proof of the asymptotic law of prime numbersQuote, 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.
Riemann Zeta Function (Wikipedia)
Logarithmic Integral Function (Wikipedia)
Includes: Logarithmic Integral Function, LeadQuote, Includes: Logarithmic Integral Function, Lead
It is relevant in problems of physics and has number theoretic significance
University of Michigan Board of Regents: Hugh L. Montgomery retirement memoir (University of Michigan Regents)
University of Michigan Board of RegentsIncludes: Hugh Montgomery, MemoirQuote, Includes: Hugh Montgomery, Memoir
He specialized in analytic number theory
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
Wikipedia: Edward Charles Titchmarsh
Includes: Edward Charles Titchmarsh, LeadQuote, Includes: Edward Charles Titchmarsh, Lead
Titchmarsh was known for work in analytic number theory
Wikipedia: Hans-Egon Richert
Includes: Hans-Egon Richert, LeadQuote, 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.
Wikipedia: Helmut Maier
Includes: Helmut Maier, LeadQuote, 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.
Vinogradov's mean value theorem (Wikipedia)
Includes: Vinogradov's Mean-Value Theorem, Lead paragraphQuote, Includes: Vinogradov's Mean-Value Theorem, Lead paragraph
It is an important inequality in analytic number theory, named for I.
Chebyshev Function (Wikipedia)
Includes: Chebyshev Function, Lead sectionQuote, 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
Jurkat-Richert theorem (Wikipedia)
Includes: Jurkat-Richert Theorem, Lead sectionQuote, 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
Z Function (Wikipedia)
Includes: Z Function, Lead sectionQuote, 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
MacTutor biography of Enrico Bombieri, St Andrews
Includes: Enrico Bombieri, SummaryQuote, 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.
Atle Selberg, Biography (MacTutor History of Mathematics)
Includes: Atle Selberg, SummaryQuote, 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.
Edmund Landau, Biography (MacTutor History of Mathematics)
Includes: Edmund Landau, SummaryQuote, 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.
Hans Rademacher, Biography (MacTutor History of Mathematics)
Includes: Hans Rademacher, SummaryQuote, 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.
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