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| Montgomery–Vaughan | |
|---|---|
| Name | Montgomery–Vaughan |
| Field | Analytic number theory |
| Statement | Distribution of primes in short intervals and mean value theorems for multiplicative functions |
| Authors | Hugh L. Montgomery; Robert C. Vaughan |
| Year | 1974 |
| Related | Bombieri–Vinogradov theorem, Large sieve, Dirichlet L-function |
Montgomery–Vaughan is a theorem in Analytic number theory established by Hugh L. Montgomery and Robert C. Vaughan that gives sharp mean-value estimates for sums over primes and multiplicative functions and yields distributional information in short intervals and arithmetic progressions. The result sits alongside classical results such as the Prime Number Theorem, the Brun–Titchmarsh theorem, and the Bombieri–Vinogradov theorem, and it has implications for the study of Dirichlet L-function zeros, the Riemann zeta function, and sieve methods like the Large sieve and the Selberg sieve.
The Montgomery–Vaughan results concern average bounds for exponential sums and mean-square estimates for weighted prime sums and multiplicative coefficients. They refine earlier work by G. H. Hardy, John E. Littlewood, and Atle Selberg and interact with major tools of Analytic number theory including estimates for character sums, the zero-free region for Dirichlet L-functions, and weighted forms of the Large sieve. Key figures whose techniques or conjectures relate include Pafnuty Chebyshev, Bernhard Riemann, Enrico Bombieri, and Alan Baker.
One standard form establishes that for a multiplicative function a(n) with suitable bounds and for X large, the mean square ∑_{q ≤ Q} ∑_{χ mod q} |∑_{n≤X} a(n) χ(n)|^2 admits a bound of the order (X + Q^2) times a small factor, paralleling the Large sieve inequality of Yakovlev and Linnik. In the prime-specific formulation, Montgomery and Vaughan proved bounds for sums ∑_{x < n ≤ x+H} Λ(n) e(nα) with Λ the von Mangoldt function that give nontrivial cancellation whenever H is larger than a threshold tied to X and the modulus. These statements often appear alongside quantifications involving the Brun–Titchmarsh theorem and estimates for exponential sums studied by I. M. Vinogradov, Nikolai Korobov, and Harold Davenport.
The theorem emerged in the 1970s amid advances in sieve theory and mean-value estimates. Montgomery and Vaughan built on the Hardy–Littlewood circle method, the Large sieve inequalities developed by Yu. V. Linnik and A. O. Gelfond, and the dispersion methods of Vaughan himself. Contemporary work by Enrico Bombieri and Andrew Granville on distribution of primes in arithmetic progressions influenced applications. Earlier milestones informing the result include the Prime Number Theorem proved by Hadamard and de la Vallée Poussin, Dirichlet's theorem on arithmetic progressions by Dirichlet, and mean-value theorems due to Ramanujan and G. H. Hardy.
Proofs combine analytic tools: the Large sieve inequality, bilinear forms estimates, Vaughan's identity decompositions, and mean-square estimates for twisted sums with Dirichlet characters. Key techniques also invoke the Fourier-analytic machinery from the Hardy–Littlewood circle method and dispersion methods developed in the context of the Bombieri–Vinogradov theorem. Montgomery and Vaughan exploited decompositions akin to Vaughan's identity and bounds for trigonometric sums related to Kloosterman sum estimates studied by André Weil and Estermann. Later expositions connect the proof to spectral methods appearing in the work of Atle Selberg and to bilinear sum techniques found in research by D. A. Burgess and Henryk Iwaniec.
The Montgomery–Vaughan bounds have been applied to problems on primes in short intervals, primes in arithmetic progressions, and multiplicative functions in mean. They inform results about gaps between primes studied by Goldston, Pintz, and Yıldırım and provide input to sieve-theoretic results by Selberg and Brun. The inequalities underpin refinements of the Brun–Titchmarsh theorem and feed into effective error terms for the Prime Number Theorem in arithmetic progressions used by Hooley and Motohashi. They also influence equidistribution results related to Weyl sums and are used in the study of moments of the Riemann zeta function by Conrey, Keating, and Snaith.
Related theorems include the Bombieri–Vinogradov theorem, the classical Large sieve, and improvements by Friedlander and Iwaniec on bilinear forms. Later generalizations adapt Montgomery–Vaughan techniques to trace functions over finite fields studied by Nick Katz and to automorphic L-functions in the program of Langlands and Iwaniec–Sarnak spectral methods. Extensions also interact with conjectures such as the Generalized Riemann Hypothesis, the Montgomery pair correlation conjecture, and the Elliott–Halberstam conjecture studied by D. A. Goldston and Timothy Gowers.
Category:Theorems in analytic number theory