In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers...

properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory). One may also study real numbers in relation...

Abstract analytic number theory is a branch of mathematics which takes the ideas and techniques of classical analytic number theory and applies them to...

Abstract analytic number theory, the application of ideas and techniques from analytic number theory to other mathematical fields Analytic combinatorics...

largest proper divisor of n can be no larger than ⁠n/2⁠. Abstract analytic number theory for information about generalizations of the theorem. Landau prime...

application of powerful analytic methods; all of these have become essential concepts and tools of modern analytic number theory. Among the new definitions...

in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, and applied mathematics, as well as in physics,...

functions and other formulas in the fields of probability, statistics, analytic number theory, and combinatorics. The gamma function can be seen as a solution...

Erdős–Kac theorem on additive functions. Number theory Analytic number theory Areas of mathematics List of number theory topics List of probability topics Probabilistic...

Multiplicative number theory is a subfield of analytic number theory that deals with prime numbers and with factorization and divisors. The focus is usually...

by either analytic methods such as Liouville's theorem, or topological ones such as the winding number, or a proof combining Galois theory and the fact...

Transcendental number theory is a branch of number theory that investigates transcendental numbers (numbers that are not solutions of any polynomial equation...

study, by the used methods, or by both. For example, analytic number theory is a subarea of number theory devoted to the use of methods of analysis for the...

analytic Eisenstein series is a special function of two variables. It is used in the representation theory of SL(2,R) and in analytic number theory....

combinatorics, geometric combinatorics, probability theory, compressed sensing and analytic number theory. Tao was born to Chinese immigrant parents and raised...

In additive number theory, the Fermat polygonal number theorem states that every positive integer is a sum of at most n n-gonal numbers. That is, every...

was a Czech-born British mathematician, working in the field of analytic number theory. He is remembered in part for the Elliott–Halberstam conjecture...

questions spurred the development of various branches of number theory, focusing on analytic or algebraic aspects of numbers. Primes are used in several...

certainly useful in applications. A large part of twentieth century analytic number theory was devoted to finding good estimates for these sums, a trend started...

June 1987) is an English mathematician working in analytic number theory and in particular the theory of prime numbers. In 2017, he was appointed Research...

generalizations. The theory of L-functions has become a very substantial, and still largely conjectural, part of contemporary analytic number theory. In it, broad...

holomorphic modular newform." Jacob Tsimerman – "For outstanding work in analytic number theory and arithmetic geometry, including breakthroughs on the André–Oort...

list of number theory topics. See also: List of recreational number theory topics Topics in cryptography Composite number Highly composite number Even and...

run time or space requirements grow as the input size grows. In analytic number theory, big O notation is often used to express a bound on the difference...

the Riemann hypothesis, is regarded as a foundational paper of analytic number theory. Through his pioneering contributions to differential geometry,...

was a Soviet mathematician, who was one of the creators of modern analytic number theory, and also a dominant figure in mathematics in the USSR. He was born...

algorithms can be found in the book Pi and the AGM – A Study in Analytic Number Theory and Computational Complexity. These two are examples of a Ramanujan–Sato...

1896 – 4 April 1981) was a German mathematician specialising in analytic number theory. He is known for, amongst other things, his contributions to the...

Johnson–Lindenstrauss lemma (Euclidean geometry) Margulis lemma Lebesgue's number lemma (dimension theory) Gauss's lemma (Riemannian geometry) Craig interpolation lemma...

But analytic number theory has had many conjectures supported by substantial numerical evidence that turned out to be false. See Skewes number for a...