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arxiv: 2605.21476 · v1 · pith:NVMVA7GWnew · submitted 2026-05-20 · 🧮 math.NT

Sharp omega results for the divisor and circle problems

Youness Lamzouri This is my paper

Pith reviewed 2026-05-21 02:42 UTC · model grok-4.3

classification 🧮 math.NT
keywords divisor problemcircle problemomega resultsresonance methodVoronoï summationGamma distributionsigned error terms
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The pith

The divisor and circle problems have error terms that attain their conjecturally sharp omega bounds with known signs.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that the error term in the number of divisors up to x and the error in counting lattice points inside a circle of radius sqrt(x) can be as large in size as the strongest conjectures allow, and it determines whether those large deviations are positive or negative. A sympathetic reader would care because these two problems test how well we understand the oscillations in basic counting functions tied to divisors and lattice points. Establishing both the maximal size and the sign together gives a sharper picture of the fluctuations than size alone. The argument introduces a resonance method that incorporates the oscillatory phase from the Voronoï summation formula directly.

Core claim

We establish omega results for the divisor and circle problems that are conjecturally sharp, while also determining the sign of the large values obtained. This improves on the work of Soundararajan and on the subsequent independent refinements of Sourmelidis and Mahatab, and gives the first improvement on Hafner's 1981 Ω+ result for the divisor problem and his Ω− result for the circle problem. The main new ingredient is a resonance method which works directly with the phase appearing in the Voronoï summation formula by replacing the usual positive kernels by a one-sided sectorial kernel, namely the density of a Gamma distribution, whose Fourier transform lies in a suitable sector of the c

What carries the argument

A resonance method that uses a one-sided sectorial kernel given by the density of a Gamma distribution to align directly with the phase in the Voronoï summation formula.

Load-bearing premise

The Fourier transform of the Gamma distribution density lies in a sector of the complex plane that is compatible with the phase of the Voronoï summation formula.

What would settle it

Numerical evaluation of the divisor error term at points x where the method predicts a large positive deviation, checking whether the observed value exceeds the conjectured lower bound in the predicted direction.

read the original abstract

We establish omega results for the divisor and circle problems that are conjecturally sharp, while also determining the sign of the large values obtained. This improves on the work of Soundararajan and on the subsequent independent refinements of Sourmelidis and Mahatab, and gives the first improvement on Hafner's 1981 $\Omega_+$ result for the divisor problem and his $\Omega_-$ result for the circle problem. The main new ingredient is a resonance method which works directly with the phase appearing in the Vorono\"i summation formula. This is achieved by replacing the usual positive kernels by a one-sided sectorial kernel, namely the density of a Gamma distribution, whose Fourier transform lies in a suitable sector of the complex plane.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript claims to establish conjecturally sharp Ω results for the divisor problem (Δ(x)) and the circle problem (P(x)), including determination of the sign of the large values attained. It improves on Soundararajan’s resonance method and the refinements of Sourmelidis and Mahatab, while providing the first improvement on Hafner’s 1981 Ω₊ result for the divisor problem and Ω₋ result for the circle problem. The central new ingredient is a resonance method applied directly to the oscillatory phase in the Voronoi summation formula, achieved by replacing positive kernels with a one-sided sectorial kernel given by the density of a Gamma distribution whose Fourier transform lies in a suitable sector of the complex plane.

Significance. If the central claims are verified, the work would constitute a meaningful advance in analytic number theory by delivering sharper omega bounds together with sign control for two classical problems. The introduction of a sectorial kernel derived from the Gamma distribution to handle the phase in the Voronoi formula represents a technical departure from earlier positive-kernel approaches and, if rigorously justified, could influence subsequent applications of resonance methods to oscillatory sums.

major comments (1)
  1. [Introduction and kernel construction] The abstract and introduction assert that the Fourier transform of the Gamma-distribution density lies in a suitable sector allowing sign control, yet the manuscript does not appear to contain an explicit computation or bound on the argument of this transform (e.g., in the section developing the kernel). Without such a calculation or a reference to a prior lemma establishing the sector condition uniformly in the relevant range, the claimed sign determination for the large values remains unverified and is load-bearing for the improvement over Hafner’s results.
minor comments (1)
  1. [Section on the new kernel] Notation for the Gamma density and its Fourier transform should be introduced with a clear definition and parameter range before its first use in the resonance argument.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comment on the kernel construction. We address the point below and will revise the paper to incorporate an explicit verification.

read point-by-point responses

  1. Referee: [Introduction and kernel construction] The abstract and introduction assert that the Fourier transform of the Gamma-distribution density lies in a suitable sector allowing sign control, yet the manuscript does not appear to contain an explicit computation or bound on the argument of this transform (e.g., in the section developing the kernel). Without such a calculation or a reference to a prior lemma establishing the sector condition uniformly in the relevant range, the claimed sign determination for the large values remains unverified and is load-bearing for the improvement over Hafner’s results.

    Authors: We agree that an explicit computation of the argument would make the sign control fully transparent and self-contained. The one-sided sectorial kernel is the density of a Gamma distribution with shape parameter α > 1, f_α(t) = t^{α−1}e^{−t}/Γ(α) for t > 0. Its Fourier transform (with the convention used in the paper) is (1 − iξ)^{−α}. For ξ real and positive, arg(1 − iξ) = −arctan(ξ), so arg((1 − iξ)^{−α}) = α arctan(ξ) lies in the open interval (0, α π/2). Choosing α sufficiently small (as is done in the resonance setup) ensures the image lies in a sector strictly inside the right half-plane, uniformly for |ξ| ≪ X^θ with θ < 1/2. We will add a short lemma (new Lemma 2.4) in the section developing the kernel that states and proves this bound, together with the resulting one-sided property. This addition confirms the sign determination without changing any of the main theorems or the comparison with Hafner’s results. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper introduces a resonance method that applies directly to the oscillatory phase in the standard Voronoï summation formula by substituting positive kernels with a one-sided sectorial kernel given by the density of a Gamma distribution. Its Fourier transform is chosen to lie in a suitable complex sector, enabling sign control for the large values while targeting conjecturally sharp Ω results. This construction is presented as an original analytic device that refines earlier positive-kernel approaches without reducing any claimed prediction or Ω result to a fitted parameter, self-referential definition, or load-bearing self-citation chain. The derivation therefore remains self-contained against external benchmarks and does not collapse by construction to its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the standard Voronoi summation formula (a known identity in analytic number theory) and introduces no new free parameters, ad-hoc constants, or postulated entities; the novelty is in the choice of kernel.

axioms (1)
  • standard math Voronoi summation formula applies to the error terms of the divisor and circle problems
    The resonance method is built directly on the phase term supplied by this classical summation formula.

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Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages · 1 internal anchor

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