Number theory
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Mathematical analysis
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Complex analysis
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History of number theory
Analytic number theory
In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav ...
Analytic number theory - Wikipedia
Riemann zeta function
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Theorems in analytic number theory
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Additive number theory
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Elliptic function
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Factorial and binomial topics
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Gamma and related functions
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Sieve theory
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Zeta and L-functions
Analytic number theory - Wikipedia
Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function, ζ(s), is a function of a complex variable s that analytically continues the sum of the infinite series which converges when the real part of s...
Riemann zeta function - Wikipedia
Theorems in analytic number theory
Additive number theory
In number theory, the specialty additive number theory studies subsets of integers and their behavior under addition. More abstractly, the field of "additive number theory" includes the study of Abeli...
Elliptic function
In complex analysis, an elliptic function is a meromorphic function that is periodic in two directions. Just as a periodic function of a real variable is defined by its values on an interval, an ellip...
Elliptic function - Wikipedia
Factorial and binomial topics
Gamma and related functions
Sieve theory
Sieve theory is a set of general techniques in number theory, designed to count, or more realistically to estimate the size of, sifted sets of integers. The primordial example of a sifted set is the s...
Zeta and L-functions
Chen's theorem
In number theory, Chen's theorem states that every sufficiently large even number can be written as the sum of either two primes, or a prime and a semiprime (the product of two primes).
The theore...
Chen's theorem - Wikipedia
Multiplicative number theory
Multiplicative number theory is a subfield of analytic number theory that deals with prime numbers and with factorization and divisors. The focus is usually on developing approximate formulas for coun...
Faà di Bruno's formula
Faà di Bruno's formula is an identity in mathematics generalizing the chain rule to higher derivatives, named after Francesco Faà di Bruno (1855, 1857), though he was not the first to state o...
Faà di Bruno's formula - Wikipedia
Analytic subgroup theorem
In mathematics, the analytic subgroup theorem is a significant result in modern transcendence theory. It may be seen as a generalisation of Baker's theorem on linear forms in logarithms.
The anal...
Bohr–Mollerup theorem
In mathematical analysis, the Bohr–Mollerup theorem is a theorem named after the Danish mathematicians Harald Bohr and Johannes Mollerup, who proved it. The theorem characterizes the gamma function, d...
Bohr–Mollerup theorem - Wikipedia
Permutation
In mathematics, the notion of permutation relates to the act of rearranging, or permuting, all the members of a set into some sequence or order (unlike combinations, which are selections of some membe...
Brun sieve
In the field of number theory, the Brun sieve (also called Brun's pure sieve) is a technique for estimating the size of "sifted sets" of positive integers which satisfy a set of conditions which are e...
Hardy–Littlewood circle method
Hardy–Littlewood circle method - Wikipedia
Rankin–Selberg method
In mathematics, the Rankin–Selberg method, introduced by (Rankin 1939) and Selberg (1940), also known as the theory of integral representations of L-functions, is a technique for directly co...
Nu function
In mathematics, the nu function iswhere is the Gamma function.
Nu function - Wikipedia
Montgomery curve
In mathematics the Montgomery curve is a form of elliptic curve, different from the usual Weierstrass form, introduced by Peter L. Montgomery in 1987. It is used for certain computations, and in parti...
Montgomery curve - Wikipedia
Maier's theorem
In number theory, Maier's theorem (Maier 1985) is a theorem about the numbers of primes in short intervals for which Cramér's probabilistic model of primes gives the wrong answer.The theorem states th...
Fundamenta nova theoriae functionum ellipticarum
In mathematics, Fundamenta nova theoriae functionum ellipticarum (new foundations of the theory of elliptic functions) is a book on Jacobi elliptic functions by Carl Gustav Jacob Jacobi. The book was...
Chebyshev's bias
In number theory, Chebyshev's bias is the phenomenon that most of the time, there are more primes of the form 4k + 3 than of the form 4k + 1, up to the same limit. This phenomenon ...
Real analytic Eisenstein series
In mathematics, the simplest real 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. It is closely rel...
Modular form
In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also...
Selberg sieve
In mathematics, in the field of number theory, the Selberg sieve is a technique for estimating the size of "sifted sets" of positive integers which satisfy a set of conditions which are expressed by c...
Montgomery's pair correlation conjecture
In mathematics, Montgomery's pair correlation conjecture is a conjecture made by Hugh Montgomery (1973) that the pair correlation between pairs of zeros of the Riemann zeta function (normaliz...
Legendre sieve
In mathematics, the Legendre sieve, named after Adrien-Marie Legendre, is the simplest method in modern sieve theory. It applies the concept of the Sieve of Eratosthenes to find upper or lower bounds...
Legendre sieve - Wikipedia
Beta function
In mathematics, the beta function, also called the Euler integral of the first kind, is a special function defined byfor The beta function was studied by Euler and Legendre and was given its name by ...
Beta function - Wikipedia
Shimizu L-function
In mathematics, the Shimizu L-function, introduced by Shimizu (1963), is a Dirichlet series associated to a totally real algebraic number field.Michael Francis Atiyah, H. Donnelly, an...