Error term improvement
TheoremDB P10194 · Candidate awaiting source review
Inferred from the community field; finer classification has not been reviewed.
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Can the exponent $1/6$ in the error term of the Bateman–Grosswald asymptotic be improved
unconditionally? That is, is there $\delta > 0$ such that
$$Q(x) = \frac{\zeta(3/2)}{\zeta(3)} x^{1/2} + \frac{\zeta(2/3)}{\zeta(2)} x^{1/3} +
O(x^{1/6 - \delta})?$$
Improvements are known under the Riemann Hypothesis.
Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review.Problem statement
Can the exponent in the error term of the Bateman–Grosswald asymptotic be improved unconditionally? That is, is there such that
O(x^{1/6 - \delta})?$$ Improvements are known under the Riemann Hypothesis.Lean 4
Formal statement
Lean 4theorem error_term_improvement :
answer(sorry) ↔ ∃ δ > (0 : ℝ),
(fun x : ℝ => (Q x : ℝ) - A * x ^ ((1 : ℝ) / 2) - B * x ^ ((1 : ℝ) / 3)) =O[atTop]
fun x => x ^ ((1 : ℝ) / 6 - δ)On-site verification needs an executable Lean target. None is attached yet.
A source attachment or link is not a verified proof. The statement’s correspondence with the original question needs human review.
Proofs & verification
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Progress & source assessment
Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.
Open. No signed Lean verification is attached to this problem or its selected resolution.
This record is in the source backlog; its novelty and unresolved status have not passed source review.
Recorded progress
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Discussion
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