Category: Theorems in analytic number theory

Kuznetsov trace formula
In analytic number theory, the Kuznetsov trace formula is an extension of the Petersson trace formula. The Kuznetsov or relative trace formula connects Kloosterman sums at a deep level with the spectr
Brun–Titchmarsh theorem
In analytic number theory, the Brun–Titchmarsh theorem, named after Viggo Brun and Edward Charles Titchmarsh, is an upper bound on the distribution of prime numbers in arithmetic progression.
Maier's theorem
In number theory, Maier's theorem is a theorem about the numbers of primes in short intervals for which Cramér's probabilistic model of primes gives a wrong answer. The theorem states that if π is the
Friedlander–Iwaniec theorem
In analytic number theory the Friedlander–Iwaniec theorem states that there are infinitely many prime numbers of the form . The first few such primes are 2, 5, 17, 37, 41, 97, 101, 137, 181, 197, 241,
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).
Bombieri–Vinogradov theorem
In mathematics, the Bombieri–Vinogradov theorem (sometimes simply called Bombieri's theorem) is a major result of analytic number theory, obtained in the mid-1960s, concerning the distribution of prim
Barban–Davenport–Halberstam theorem
In mathematics, the Barban–Davenport–Halberstam theorem is a statement about the distribution of prime numbers in an arithmetic progression. It is known that in the long run primes are distributed equ
Riemann–von Mangoldt formula
In mathematics, the Riemann–von Mangoldt formula, named for Bernhard Riemann and Hans Carl Friedrich von Mangoldt, describes the distribution of the zeros of the Riemann zeta function. The formula sta
Petersson trace formula
In analytic number theory, the Petersson trace formula is a kind of orthogonality relation between coefficients of a holomorphic modular form. It is a specialization of the more general Kuznetsov trac
Ramanujan's master theorem
In mathematics, Ramanujan's Master Theorem, named after Srinivasa Ramanujan, is a technique that provides an analytic expression for the Mellin transform of an analytic function. The result is stated
Riemann–Siegel formula
In mathematics, the Riemann–Siegel formula is an asymptotic formula for the error of the approximate functional equation of the Riemann zeta function, an approximation of the zeta function by a sum of
Vaughan's identity
In mathematics and analytic number theory, Vaughan's identity is an identity found by R. C. Vaughan that can be used to simplify Vinogradov's work on trigonometric sums. It can be used to estimate sum
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 was first observed b
Landsberg–Schaar relation
In number theory and harmonic analysis, the Landsberg–Schaar relation (or identity) is the following equation, which is valid for arbitrary positive integers p and q: The standard way to prove it is t
Fundamental lemma of sieve theory
In number theory, the fundamental lemma of sieve theory is any of several results that systematize the process of applying sieve methods to particular problems. Halberstam & Richertwrite: A curious fe
Landau prime ideal theorem
In algebraic number theory, the prime ideal theorem is the number field generalization of the prime number theorem. It provides an asymptotic formula for counting the number of prime ideals of a numbe
Jurkat–Richert theorem
The Jurkat–Richert theorem is a mathematical theorem in sieve theory. It is a key ingredient in proofs of Chen's theorem on Goldbach's conjecture.It was proved in 1965 by Wolfgang B. Jurkat and Hans-E
Prime number theorem
In mathematics, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common a
Linnik's theorem
Linnik's theorem in analytic number theory answers a natural question after Dirichlet's theorem on arithmetic progressions. It asserts that there exist positive c and L such that, if we denote p(a,d)
Perron's formula
In mathematics, and more particularly in analytic number theory, Perron's formula is a formula due to Oskar Perron to calculate the sum of an arithmetic function, by means of an inverse Mellin transfo
Kronecker limit formula
In mathematics, the classical Kronecker limit formula describes the constant term at s = 1 of a real analytic Eisenstein series (or Epstein zeta function) in terms of the Dedekind eta function. There
Siegel–Walfisz theorem
In analytic number theory, the Siegel–Walfisz theorem was obtained by Arnold Walfisz as an application of a theorem by Carl Ludwig Siegel to primes in arithmetic progressions. It is a refinement both
Vinogradov's mean-value theorem
In mathematics, Vinogradov's mean value theorem is an estimate for the number of equal sums of powers.It is an important inequality in analytic number theory, named for I. M. Vinogradov. More specific
Hardy–Ramanujan theorem
In mathematics, the Hardy–Ramanujan theorem, proved by G. H. Hardy and Srinivasa Ramanujan, states that the normal order of the number ω(n) of distinct prime factors of a number n is log(log(n)). Roug