Elliott–Halberstam conjecture

The Elliott–Halberstam conjecture predicts that primes are distributed nearly uniformly among reduced residue classes on average over moduli up to nearly the square of the prime-counting range.

Prime number theorem for arithmetic progressions: A theorem describing the asymptotic distribution of primes among reduced residue classes for each fixed modulus. The conjecture extends this distributional picture by averaging errors across many moduli.
Peter D. T. A. Elliott: A British mathematician known for contributions to analytic number theory and sieve methods. His work with Halberstam gave the conjecture its first half of its name.
Bounded gaps between primes: The phenomenon that infinitely many pairs of distinct primes differ by at most a fixed constant. Elliott–Halberstam assumptions can sharpen sieve bounds for the gap size.
Generalized Riemann hypothesis: A conjecture that nontrivial zeros of Dirichlet L-functions lie on the critical line. It offers a different route to prime-distribution estimates, but does not simply assert this averaged level.
Von Mangoldt function: An arithmetic function that equals log p on prime powers and zero otherwise. Its weighted sums encode prime counts in the conjecture’s error terms.
Heini Halberstam: A British mathematician whose research included sieve methods and the distribution of primes. His joint work with Elliott formulated the conjectural distribution bound.
Maynard–Tao theorem: A theorem proving that infinitely many intervals of fixed length contain at least a specified number of primes. Its sieve framework gains stronger numerical consequences from distribution hypotheses such as this conjecture.
Parity problem: A limitation of sieve methods that prevents them from distinguishing reliably between numbers with odd and even prime-factor counts. Even near-optimal distribution estimates do not automatically remove this central obstacle to sieve proofs.
Reduced residue system: The residue classes modulo an integer that are coprime to that integer. The conjecture averages prime distribution across precisely these classes.
Sieve theory: A collection of methods for estimating integers that avoid specified divisibility conditions. The conjecture’s motivation and many consequences are expressed through sieve estimates.

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Hardy–Littlewood conjecturesRelated: Stronger distribution estimates of this kind would support progress on prime-pattern questions.