Elliott–Halberstam conjecture

Elliott–Halberstam conjecture is a conjecture in analytic number theory about the distribution of prime numbers in arithmetic progressions that strengthens the Bombieri–Vinogradov theorem and has deep implications for problems such as bounded gaps between twin primes and patterns studied in the Prime k-tuples conjecture. It was proposed by Peter D. T. A. Elliott and Heini Halberstam and influenced later work by researchers associated with Yitang Zhang, Terence Tao, and the Polymath Project; its resolution would affect results connected to the Goldbach conjecture, Hardy-Littlewood conjectures, and sieves developed by Enrico Bombieri and Atle Selberg.

Statement of the conjecture

The conjecture asserts that for every positive ε and every A>0 there exists a constant depending on ε and A such that the sum over moduli q up to about x^{1-ε} of the maximal deviation of π(x;q,a) from the expected x/φ(q) is bounded by x/(log x)^A; here π(x;q,a) counts primes ≤x congruent to a modulo q relative to Euler's φ function, drawing on notions used by Paul T. Bateman, Roger Heath-Brown, D. R. Heath-Brown, H. L. Montgomery, and techniques from work of Atle Selberg and Enrico Bombieri. The conjecture refines uniformity ranges in estimates originally established in the Bombieri–Vinogradov theorem and connects to distributional hypotheses considered by G. H. Hardy and John Edensor Littlewood in their conjectures on primes in progressions and correlations.

Historical background and motivation

Elliott and Halberstam formulated their conjecture in the late 1960s inspired by the limits of the Bombieri–Vinogradov theorem and earlier large-sieve inequalities due to figures such as Yuri Linnik, H. Iwaniec, and Kurt Mahler; contemporaneous influence came from sieve methods of Brun and later refinements by Alfréd Rényi and Atle Selberg. Motivations included sharpening results toward the Twin prime conjecture and the Hardy–Littlewood prime k-tuples conjecture as pursued by G. H. Hardy, J. E. Littlewood, and later computational explorations by Oliveira e Silva and collaborators. The conjecture became a focal point for collaborative efforts epitomized by the Polymath Project and drove work culminating in breakthroughs by Yitang Zhang and follow-up refinements by James Maynard and Terence Tao.

Known results and partial progress

Partial progress includes the unconditional Bombieri–Vinogradov theorem attributable to Enrico Bombieri and A. I. Vinogradov and improvements using the Large sieve developed by Patrick X. Gallagher and Henryk Iwaniec. Zhang's bounded gaps theorem, using ideas from Goldston–Pintz–Yıldırım and Yitang Zhang, relied on weaker distributional estimates that fall short of the conjecture but nonetheless produced finite prime gaps; subsequent optimizations by Polymath8 and James Maynard tightened bounds. Results conditional on variants like the Generalized Riemann Hypothesis and the Montgomery–Vaughan conjecture show how assuming Elliott–Halberstam yields dramatic consequences for prime constellations and the Twin prime conjecture; analytic techniques from Iwaniec–Kowalski and spectral methods inspired by Atle Selberg have been marshaled in partial verifications in specialized ranges and averaged forms.

Consequences and applications

Assuming the conjecture leads to bounded gaps between primes and would imply the existence of infinitely many prime pairs within bounded intervals as predicted by work of D. A. Goldston, J. Pintz, and C. Y. Yıldırım; it strengthens predictions of the Hardy–Littlewood conjectures concerning prime k-tuples and refines estimates relevant to the Goldbach conjecture. In computational number theory contexts explored by Andrew Granville, K. Soundararajan, and Terence Tao, Elliott–Halberstam would improve error terms for prime distribution in arithmetic progressions and hence inform heuristic models used in algorithms by researchers at institutions like Massachusetts Institute of Technology and Princeton University. Connections extend to multiplicative number theory results of Elliott himself, sieve-theoretic frameworks of Atle Selberg, and implications for equidistribution questions related to Dirichlet characters and L-functions studied by Hugh Montgomery and Andrew Odlyzko.

Related conjectures and refinements

Several related hypotheses include the generalized Elliott–Halberstam statements proposed in collaborative projects such as Polymath8, the Generalized Riemann Hypothesis linking zeros of Dirichlet L-functions as formulated in extensions of work by Bernhard Riemann and Atle Selberg, and the Montgomery pair correlation conjectures advanced by Hugh Montgomery and examined by Freeman Dyson. Variants like the Motohashi formula applications by Y. Motohashi and weighted forms considered in the Goldston–Graham–Pintz–Yıldırım framework show structural similarities; refinements also interact with conjectures on prime gaps pursued by James Maynard and conditional results leveraging hypotheses due to Montgomery–Vaughan.

Category:Conjectures in number theory