Elliott–Halberstam conjecture

The Elliott–Halberstam conjecture: for every θ < 1 and A > 0 there exists a constant C > 0 such that Σ_1 ≤ q ≤ x^θ E(x; q) ≤ C x/log^A x for all x > 2.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-elliott-halberstam-conjecture,
  title        = {Elliott–Halberstam conjecture},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/elliott-halberstam-conjecture}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

The Elliott–Halberstam conjecture: for every $θ < 1$ and $A > 0$ there exists a constant $C > 0$ such that $1 \leq q \leq x^{θ} \sum E (x; q) \leq \frac{C x}{lo g ^{A} x}$ for all $x > 2$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Wikipedia.ElliottHalberstamConjecture.

theorem elliott_halberstam (θ : ℝ) (hθ : θ < 1) (A : ℝ) (hA : 0 < A) :
    ∃ C > (0 : ℝ), ∀ x : ℕ, 2 < x →
      ∑ q ∈ Finset.Icc 1 ⌊(x : ℝ) ^ θ⌋₊, E x q ≤ C * x / Real.log x ^ A

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

Wikipedia
[EH68] Elliott, Peter D. T. A. and Halberstam, Heini, *A conjecture in prime number theory*, Symposia Mathematica, Vol. IV (INDAM, Rome, 1968/69), 59–72.
[FG89] Friedlander, John and Granville, Andrew, *Limitations to the equi-distribution of primes I*, Ann. of Math. (2) 129 (1989), no. 2, 363–382.

Source and licence

Imported from Formal Conjectures (Wikipedia), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.