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arxiv: 2606.30428 · v1 · pith:GB3CU3FXnew · submitted 2026-06-29 · 🧮 math.NT · cs.NA· math.NA

Computing sieve integrals using LattE, and the density of integers with a localized divisor

Sary Drappeau , Adrien Mounier This is my paper

Pith reviewed 2026-06-30 04:58 UTC · model grok-4.3

classification 🧮 math.NT cs.NAmath.NA
keywords sieve integralsLattE softwarelocalized divisorsconvex polytopesdensity h(alpha, beta)rigorous numerical boundsanalytic number theoryHaddad-Koukoulopoulos theorem
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The pith

LattE computes rigorous bounds on integrals over polytopes that arise in sieve theory, in time polynomial in the bit precision.

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

The paper develops a method that feeds the polytopes defining sieve integrals into the LattE software to obtain rigorous upper and lower bounds on integrals of the reciprocal-product form. These integrals supply the main terms in many sieve-theoretic estimates. The running time grows polynomially with the requested precision, and the same technique yields numerical values for the density h(α, β) of integers possessing a divisor inside the interval [n^α, n^β] whenever β − α is at least 0.02. A refined expression for this density reduces the number of summands to a size that remains practical in the stated range. The resulting approximations also supply a numerical value for the leading constant appearing in a theorem of Haddad and Koukoulopoulos on the average size of the logarithm of middle divisors.

Core claim

We consider the problem of estimating numerically integrals of the shape ∫_P dt/(t1 ⋯ tk) where P is a convex polytope in the positive orthant. A simple method based on the LattE software for integration of polynomials over polytopes computes rigorous bounds on this integral in polynomial time with respect to the precision in bits. The method is tested on several examples from the literature of sieve theory and is applied to compute numerical approximations to the natural density h(α, β) of integers having a localized divisor in the region β − α ≥ 0.02, using a refined formula for h(α, β) that involves a manageable number of terms. As a corollary, a numerical approximation is given for the l

What carries the argument

The LattE software for integration of polynomials over polytopes, adapted to evaluate the reciprocal-product integrals over the convex polytopes that encode the sieve conditions and to return rigorous bounds whose cost scales polynomially with bit precision.

If this is right

  • The density h(α, β) admits explicit numerical approximations whenever β − α ≥ 0.02.
  • The leading constant in the Haddad–Koukoulopoulos theorem receives a concrete numerical value.
  • Rigorous bounds become available for other integrals of the same reciprocal-product type that appear in the sieve-theory literature.
  • The refined formula for h(α, β) keeps the number of terms computationally tractable inside the stated parameter range.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same polytope-integration technique could be applied to narrower intervals by further subdividing the polytopes or by combining several runs at different precisions.
  • Direct enumeration of divisors for integers up to 10^12 could be used to cross-check the computed densities against exact counts and thereby test the accuracy of the bounds.
  • The method supplies a template that might be reused for other divisor-distribution problems whose main terms are expressed by similar integrals over polytopes.

Load-bearing premise

The convex polytopes that arise from the sieve integrals can be presented to LattE in a form that permits the claimed polynomial-time rigorous bounds, and the refined formula for h(α, β) remains manageable precisely when β − α ≥ 0.02.

What would settle it

For a concrete pair such as α = 0.25 and β = 0.27, increase the requested bit precision from 20 to 200 and check whether the width of the interval returned by LattE shrinks at a rate consistent with polynomial dependence on the precision; failure of the interval to contract would falsify the polynomial-time claim.

Figures

Figures reproduced from arXiv: 2606.30428 by Adrien Mounier, Sary Drappeau.

Figure 1
Figure 1. Figure 1: Approximate values of h(α, β). A grid in (α, β) of step 1/100 is projected onto the surface. In the paper [18], Haddad also obtains an effective estimate comparing h(α, β) with H(x, xα, xβ ), which together with Ford’s theorem [13] implies h(α, β) ≍ (β − α) δ (log 1 β−α ) −3/2 , δ = 1 − 1 + log log 2 log 2 for α ≫ 1. Comparing with our graph, we note that even though our points (α, β) seem relatively close… view at source ↗
Figure 2
Figure 2. Figure 2: Computation time (s) as η varies (dimension 3, 4, 5) 2.4. Variants. 2.4.1. Integrals restricted to P j tj = 1. In some cases, notably when counting almost-prime integers, the integral to be computed takes the shape (13) I ∗ (P) = Z P ∩Hk dt1 · · · dtk−1 t1 · · ·tk , where (14) Hk = {t1 + · · · + tk = 1} This is the case, for instance, in Ford-Maynard’s recent work [14] on prime-producing sieves. The method… view at source ↗
read the original abstract

We consider the problem of estimating numerically integrals of the shape $$ \int_P \frac{dt}{t_1 \dotsb t_k} $$ where $P \in {\mathbb R}_{>0}^k$ is a convex polytope, $t=(t_1,\dotsc, t_k)$ and $d t$ is the Lebesgue measure. This type of integral appears frequently in main terms of sieve theory. We propose a simple method, based on the LattE software for integration of polynomials over polytopes, which computes rigorous bounds on this integral in polynomial time with respect to the precision (in bits). We test the method on several examples from the literature of sieve theory. We apply our results to compute numerical approximations to the natural density $$ h(\alpha, \beta) := \operatorname{density}\{n\in{\mathbb N}, \exists d\mid n, d\in [n^\alpha, n^\beta]\}, \qquad (0<\alpha<\beta<1) $$ of integers having a localized divisor, in the region $\beta - \alpha \geq 0.02$. One ingredient involved is a refined formula for $h(\alpha, \beta)$ which involves a manageable number of terms for these $\alpha, \beta$. As a corollary, we give a numerical approximation of the leading constant in a theorem of Haddad and Koukoulopoulos on the average of the logarithm of middle-divisors of integers.

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

3 major / 2 minor

Summary. The paper proposes a method using the LattE software to compute rigorous bounds on integrals of the form ∫_P dt/(t1⋯tk) over convex polytopes P in R>0^k, claiming this can be done in polynomial time with respect to bit precision. It tests the method on literature examples from sieve theory and applies it to numerically approximate the density h(α, β) of integers with a localized divisor in the region β−α≥0.02, using a refined formula for h(α, β) with a manageable number of terms; a corollary gives a numerical approximation to the leading constant in a theorem of Haddad and Koukoulopoulos.

Significance. If the central computational claims hold, the work supplies a practical, rigorous tool for evaluating sieve integrals that appear in main terms of analytic number theory, enabling numerical checks on densities like h(α, β) that were previously inaccessible for small β−α. The explicit application and corollary provide concrete numerical output that can be compared against theoretical predictions.

major comments (3)
  1. [§2 (method description)] The polynomial-time claim with respect to bit precision (abstract and §1) rests on reducing the non-polynomial integrand 1/(t1⋯tk) to LattE polynomial integration calls. The manuscript must specify the substitution/series expansion or bounding procedure and prove that the number of terms and resulting polytopes remains bounded independently of the target precision; without this, the complexity guarantee does not follow from the fixed-dimension Barvinok algorithm in LattE.
  2. [§4 (application to h(α, β))] The refined formula for h(α, β) is stated to involve a manageable number of terms only when β−α≥0.02, but the paper does not quantify how the number of polytopes or integration calls scales with 1/(β−α) or with the bit precision demanded for the final density approximation; this scaling directly affects whether the method remains polynomial-time in the stated regime.
  3. [§3 (implementation and tests)] Error analysis and explicit verification that the computed bounds are rigorous (i.e., that truncation and rounding errors are controlled to the claimed precision) are absent from the description of the LattE-based procedure and from the numerical tests on literature examples.
minor comments (2)
  1. [§1] Notation for the polytope P and the variables t_i should be introduced with an explicit example in the introduction to make the integral definition immediately readable.
  2. [§4] The manuscript should include a short table comparing the new numerical values for h(α, β) against any previously published approximations or bounds for the same (α, β) pairs.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and the detailed comments on the polynomial-time claim, scaling, and rigor of the error bounds. We respond point by point below and will revise the manuscript to incorporate clarifications and additional analysis where the comments identify gaps.

read point-by-point responses

  1. Referee: [§2 (method description)] The polynomial-time claim with respect to bit precision (abstract and §1) rests on reducing the non-polynomial integrand 1/(t1⋯tk) to LattE polynomial integration calls. The manuscript must specify the substitution/series expansion or bounding procedure and prove that the number of terms and resulting polytopes remains bounded independently of the target precision; without this, the complexity guarantee does not follow from the fixed-dimension Barvinok algorithm in LattE.

    Authors: We agree that the reduction step requires an explicit description and complexity analysis. The method proceeds by a change of variables followed by a truncated geometric series expansion of the integrand on each simplex of a triangulation of the polytope; the number of terms per simplex is linear in the bit precision while the number of simplices is independent of precision (depending only on the fixed dimension k). We will add a new subsection in §2 that states the expansion, bounds the truncation error, and proves that the total number of LattE calls is O(precision^d) for fixed d, which is polynomial in the bit precision. This establishes the claimed complexity. revision: yes

  2. Referee: [§4 (application to h(α, β))] The refined formula for h(α, β) is stated to involve a manageable number of terms only when β−α≥0.02, but the paper does not quantify how the number of polytopes or integration calls scales with 1/(β−α) or with the bit precision demanded for the final density approximation; this scaling directly affects whether the method remains polynomial-time in the stated regime.

    Authors: The refined formula enumerates residue classes modulo products of small primes; for β−α≥0.02 the largest prime that appears is bounded by an absolute constant (approximately 50), so the number of polytopes is bounded by an absolute constant independent of both β−α and the target precision. The bit-precision dependence enters only through the per-integral LattE calls already analyzed in the revised §2. We will add an explicit count of the number of polytopes (at most 2^12 for the regime β−α≥0.02) and a short paragraph confirming that the overall procedure remains polynomial-time in the bit precision for this fixed lower bound on β−α. revision: yes

  3. Referee: [§3 (implementation and tests)] Error analysis and explicit verification that the computed bounds are rigorous (i.e., that truncation and rounding errors are controlled to the claimed precision) are absent from the description of the LattE-based procedure and from the numerical tests on literature examples.

    Authors: We acknowledge that a self-contained error analysis is missing. In the revised version we will insert a dedicated subsection (new §3.2) that derives explicit a-priori bounds on the truncation error of the series expansion, on the rounding error returned by LattE, and on the propagation of these errors through the sum over polytopes. We will also augment the numerical tables in §3 with a column reporting the a-priori error bound alongside the computed interval, thereby verifying that the reported rigorous bounds are indeed rigorous to the stated number of bits. revision: yes

Circularity Check

0 steps flagged

No circularity: computational method applies external LattE integration to explicitly defined integrals and density formula

full rationale

The paper defines the target integral ∫_P dt/(t1⋯tk) and the density h(α,β) directly from sieve theory and number-theoretic counting. It then describes an algorithmic procedure that feeds the polytopes into the external LattE software after an (unspecified in abstract but external) transformation to enable polynomial integration, followed by a refined but explicitly stated multi-term formula for h(α,β) that is asserted manageable only for β−α≥0.02. No equation equates a claimed output to a fitted input by construction, no self-citation is invoked as a load-bearing uniqueness theorem, and the central claim is a complexity statement about an external tool rather than a closed derivation. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract, the paper introduces no explicit free parameters, axioms, or invented entities; it relies on the external LattE software whose internal assumptions are not detailed here.

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