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| Montgomery–Vaughan conjecture | |
|---|---|
| Name | Montgomery–Vaughan conjecture |
| Field | Number theory |
| Proposer | Hugh L. Montgomery; Robert C. Vaughan |
| Year | 1974 |
| Related | Goldbach conjecture; twin prime conjecture; Selberg sieve; large sieve; Bombieri–Vinogradov theorem |
Montgomery–Vaughan conjecture The Montgomery–Vaughan conjecture is a quantitative hypothesis in analytic Number theory about the distribution of multiplicative functions, particularly the size of exponential sums and mean values tied to primes and divisor functions. It connects work of Hugh L. Montgomery and Robert C. Vaughan with major results of Enrico Bombieri, Atle Selberg, and Peter Sarnak, and it sits alongside central problems such as the Goldbach conjecture and the twin prime conjecture in guiding methods for primes in arithmetic progressions and correlations of arithmetic functions.
The conjecture asserts a specific upper bound for mean-square sums of multiplicative functions over short intervals or arithmetic progressions: roughly, for a well-behaved multiplicative function f (notably the von Mangoldt function Λ or the divisor function d_k) and for parameters X, H with H growing modestly with X, the second moment ∑_{x < n ≤ x+H} f(n) has variance bounded by the classical square-root cancellation prediction up to logarithmic factors. Montgomery and Vaughan formulated the bound in terms of exponential sums and bilinear forms, predicting that for large X and H = X^θ with 0<θ<1 the mean-square is ≪ H log^C X for an explicit constant C. Their statement ties to bounds for sums ∑_{n ≤ N} a_n e(nα) where a_n are arithmetic coefficients and α is real, and to conjectured optimal ranges for the Large sieve and Bombieri–Vinogradov theorem-type uniformity in moduli.
Montgomery and Vaughan proposed their conjecture during work on mean values and exceptional sets in the 1970s, building on earlier techniques developed by Atle Selberg, Harold Davenport, and I. M. Vinogradov. The conjecture emerged from attempts to quantify error terms in the Prime Number Theorem in arithmetic progressions and from refinements of the Large sieve inequality by Enrico Bombieri and H. L. Montgomery. Subsequent interactions involved contributions by Alan Baker, G. H. Hardy, John Littlewood, and later analytic number theorists such as Roger Heath-Brown, Henryk Iwaniec, and K. Soundararajan who explored related exponential sum bounds. The Montgomery–Vaughan conjecture became a touchstone influencing approaches to the Goldbach conjecture and to mean-value theorems for multiplicative functions.
Work by Montgomery and Vaughan themselves produced several conditional and unconditional bounds that approach the conjectured strength; for instance, their large sieve refinements and bilinear form estimates yield nontrivial upper bounds for mean squares of coefficients like Λ and d_k. The Bombieri–Vinogradov theorem offers average results over moduli that align with Montgomery–Vaughan predictions for certain ranges of parameters, and the Barban–Davenport–Halberstam theorem gives complementary mean-square control in other regimes. Results by Roger Heath-Brown on divisor correlations, and by D. R. Heath-Brown and J. Friedlander on bilinear sums, provide partial confirmations for specific functions or restricted ranges of H. Conditional improvements assuming the Generalized Riemann Hypothesis or conjectures on zeros of L-functions such as those studied by Atle Selberg and P. X. Gallagher give sharper forms of the conjectured bounds.
Approaches to the conjecture deploy a toolkit of analytic methods: the large sieve of Enrico Bombieri and H. L. Montgomery, the dispersion method of H. L. Montgomery and Vaughan, bilinear form decompositions, weighted exponential sum estimates, and spectral techniques related to automorphic forms studied by Atle Selberg and Peter Sarnak. Further techniques include the use of sieve methods as developed by Heath-Brown and J. Friedlander, shifted convolution sums tied to the theory of Modular forms and Maass forms, and estimates for moments of L-functions derived from work of Hugh Montgomery and Andrew Granville. Harmonic analysis on Adeles and trace formulas originating in the work of Selberg and James Arthur also inform attempts to access uniformity in moduli and short-interval behavior.
Numerical investigations have tested special cases: computations for sums of the von Mangoldt function Λ over short intervals or arithmetic progressions compare observed variances with Montgomery–Vaughan predictions, with data produced by computational projects influenced by methods used in studies of the Riemann zeta function and zeros examined by Andrew Odlyzko. For divisor functions d_k, extensive numerical work by researchers building on algorithms from Richard Brent and John Pollard shows compatible behavior in ranges accessible to computation, while explicit counterexamples have not been found. Empirical support is strongest for moderate ranges of X and H where sieve and spectral methods predict the dominant terms.
Key open problems include proving the full conjecture for Λ and d_k, extending Bombieri–Vinogradov type uniformity to the conjectured ranges, and connecting the conjecture to deep properties of zeros of L-functions studied in the context of the Generalized Riemann Hypothesis and the Montgomery pair correlation conjecture. Progress may come from advances in trace formula techniques by researchers influenced by James Arthur and Peter Sarnak, novel bilinear sum estimates by analysts in the tradition of Montgomery and Vaughan, or breakthroughs in sieve methods inspired by Goldston, Pintz, and Yıldırım. The conjecture remains a central guiding estimate in contemporary analytic Number theory research, shaping both theoretical development and computational exploration.