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Convolution-type Bombieri-Vinogradov theorem with well-factorable Weights, and its applications

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Pith's one-line read A new convolution-type Bombieri–Vinogradov theorem shows primes weighted by smooth factors stay equidistributed to moduli of size about x^{4/7}, and yields density >0.296 for the consecutive largest-prime-factor problem and a sharper…

desk verdict A serious convolution-type Bombieri–Vinogradov paper with real record improvements in two applications, but the referee should spend most of their time on the unverified transfer of BFI and Pascadi hypotheses to rough-number sequences. read the letter →

arxiv 2608.13299 v1 pith:SNAY63KG submitted 2026-08-13 math.NT

Zhiyuan Yang This is my paper · ORCID
classification math.NT MSC 11N1311N2511N3611N37
keywords primesinarithmeticprogressionsBombieri–Vinogradovtheoremwell-factorableweightslinearsievelargestprimefactorconsecutiveintegersshiftedDickmanfunction
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper sets out to prove a convolution-type analogue of the classical equidistribution theorem for primes in arithmetic progressions: instead of testing a prime alone, test a prime multiplied by an integer $l\sim x^\nu$ lying near a fixed scale, and ask for the same level of distribution with well-factorable weights. The claimed exponent $L(\nu)$ is piecewise-linear and never drops below $4/7$; for the upper-bound well-factorable linear sieve weights it is at least $3/5$, and a two-parameter version allows an additional divisor-bounded factor $d\sim x^\theta$. If these estimates are correct, they immediately give two results: more than $0.296x$ of integers $n\le x$ satisfy $P^+(n)0.7404$. These improve the previous $0.280$ density bound and the previous $\frac72\log\frac1c$ shifted-prime bound, pushing two long-standing problems about largest prime factors closer to their conjectured answers.

What carries the argument

The central object is the well-factorable weight: a bounded sequence $\lambda_q$ of level $Q$ such that for every split $Q_1Q_2=Q$ one can factor $\lambda=\alpha*\beta$ with the two factors supported on $q_1\le Q_1$ and $q_2\le Q_2$. This flexibility lets the proof redistribute the modulus into pieces while estimating exponential sums. Around it the proof builds a triply-well-factorable convolution estimate (a weight that factors into three bounded pieces), an exceptional large sieve assumption, and bounds for incomplete Kloosterman sums; these produce the averaging lemmas used in the core cases. The decomposition $l=l_0l_1$ splits the smooth factor $l_0$, removed by the Dickman–de Bruijn estimate, from the large-prime-factor part $l_1$, and the upper-bound linear sieve weights together with a factorization proposition supply the $3/5$ level used in the applications.

What would settle it

Recompute the numerical integrals defining $C(c,\delta_1)$ at $c=0.1348$, $\delta_1=0.417$: if $C(c,\delta_1)-2\nu\le0.296$ for $\nu=10^{-6}$, Theorem 1.3 fails as stated. Independently, construct a tuple of the form in Lemma 2.5 with $q\asymp L^2$ and $N,H$ as specified, and numerically evaluate the exceptional Maass-form sum in Assumption 2.3; one tuple exceeding the quoted bound would cut the $4/7$ range in Case 2.

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Extended reading notes

Core claim

The paper's own claim, stated as Theorems 1.1 and 1.2, is that the convolution-type discrepancy (1.3) is bounded by $O_A(x/(\log x)^A)$ for well-factorable weights of level $x^{L(\nu)-\varepsilon}$, with $L(\nu)$ taking the listed piecewise values $\frac58-\frac{189}{200}\nu$, $\frac47$, $\frac12+\frac{\nu}{2}$, $1-\frac52\nu$, $\frac12+\frac{\nu}{3}$, $\frac{32}{39}-\frac{34}{39}\nu$, $\frac25+\frac35\nu$, $\frac14+\nu$, and $\frac12+\frac{\nu}{2}$ on successive intervals of $\nu$; for upper-bound linear sieve weights the admissible level is at least $x^{3/5-\varepsilon}$ in the range $\nu\le1/3$ and agrees with $L(\nu)$ beyond it. Theorem 1.3 then asserts that $\#\{n\le x:P^+(n)<P^+(n+1)\}>0.296x$ for large $x$, and Theorem 1.4 asserts that $\limsup_{x\to\infty}T_c(x)/\pi(x)\le S(c)$, with $S(c)$ given by an explicit integral and $S(c)<\frac72\log\frac1c$ for $0.7404<c<1$; in particular the range where the $\frac72\log\frac1c$ bound is valid improves from $c>e^{-2/7}\approx0.7515$ to $c>0.7404$.

Load-bearing premise

The load-bearing premise is that two imported tools—the exceptional large sieve bound of Assumption 2.3 with the stated $Y$-parameters, and the entirety of the classical $4/7$ method's technical assumptions applied to the new convolution weights—hold exactly as quoted; if either fails for a hidden edge case, the exponents and both application constants do not follow.

Editorial extensions

If this is right

  • For every $0<\nu\le1$, the convolution form of the equidistribution estimate holds at least to moduli of size $x^{4/7-\varepsilon}$, so adding the smooth factor $l\sim x^\nu$ costs no distribution range.
  • With upper-bound well-factorable linear sieve weights, the distribution level reaches $x^{3/5-\varepsilon}$ for $\nu\le1/3$, letting sieve arguments that require non-negative weights operate at the higher $3/5$ exponent.
  • The consecutive-integer pattern $P^+(n)<P^+(n+1)$ occurs for more than $29.6\%$ of integers, and the same density bound holds for the reverse pattern, improving the previous $28\%$ bound.
  • For shifted primes, the upper density of $p$ with $P^+(p-1)\ge p^c$ is below $\frac72\log\frac1c$ for every $c>0.7404$, improving the previous validity range $c>e^{-2/7}$; for $c>0.8602$ that density is below $1/2$.
  • The two-parameter estimate at level $L(\theta,\nu)$ makes the distribution theorem available inside nested sums over $d$ and $q$, the exact shape needed in convolution problems and further sieve arguments.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same weighted-discrepancy mechanism should transfer to other shifted patterns such as $P^+(n)<P^+(n+k)$ for fixed $k$, where the obstruction is the same averaged error term; a $k$-shifted analogue is a direct testable extension.
  • The paper's closing remark points to replacing an exponent $1/2$ by $4/7$ in one estimating step, which suggests the constant can be nudged from $0.296$ toward $0.297$ with the present method but not toward the conjectured $1/2$ density.
  • Because the exceptional large sieve Assumption 2.3 is imported rather than proved, the four theorems are conditional on that hypothesis in its stated generality; a proof of the assumption for the sequences appearing in Lemma 2.5 would make all the applications unconditional at once.
  • The formula for $S(c)$ depends only on the upper-bound sieve level $L'(u)$, not on the full convolution exponent, so any future improvement of $L'(u)$ at small $u$ translates directly into lower values of $S(c)$.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper proves a convolution-type Bombieri-Vinogradov theorem (Theorem 1.1) for sequences weighted by well-factorable functions after splitting off a rough-number component, with exponent L(nu) at least 4/7, and an analogous two-parameter statement (Theorem 1.2) with exponent L(theta,nu). The proof follows the Bombieri-Friedlander-Iwaniec (BFI) strategy, replacing the prime indicator by the rough-number indicator beta_n = 1_{l1 in L1} 1_{P^-(l1)>z} and invoking Pascadi's exceptional large sieve estimates (Lemmas 2.5, 2.6). As applications, Theorem 1.3 claims #{n<x: P^+(n)<P^+(n+1)} > 0.296 x, improving the author's earlier 0.280, and Theorem 1.4 claims limsup T_c(x)/pi(x) <= S(c) < (7/2) log(1/c) for 0.7404<c<1, improving the Ding-Wang bound. The applications depend on the upper-bound linear-sieve variant L'(nu), with L'(nu) >= 3/5.

Significance. If the main theorem and its transfer to the rough-number sequence are correct, this is a substantial new distribution estimate for primes in arithmetic progressions with convolution weights, and it yields new record constants in two longstanding problems on largest prime factors. The paper gives detailed case analyses following BFI and makes explicit use of recent tools of Pascadi, which is a meritorious feature. The significance is conditional, however, on two load-bearing points that are currently not proved in the manuscript: the inheritability of all BFI hypotheses by the rough-number indicator, and the numerical validation of the claimed constant 0.296.

major comments (3)
  1. [Section 3.2 (and Section 3.1, application of Lemma 3.1)] The proof of Theorem 1.1 replaces the BFI prime sequence by beta_n = 1_{l1 in L1} 1_{P^-(l1)>z} and asserts in Section 3.2 that 'all assumptions (A_i) in [2] are satisfied whenever needed.' No verification is provided for any of the BFI hypotheses, in particular the Siegel-Walfisz condition, the Type I/II estimates, or the exponential-sum conditions, for this rough-number indicator. The same silent transfer occurs in Section 3.1 when Lemma 3.1 is applied with alpha_n set to 1_{l1 in L1} 1_{P^-(l1)>z} and the Siegel-Walfisz condition is taken for granted. The levels L(nu) and L'(nu) in Theorems 1.1-1.4 inherit any loss from these hypotheses, so this is not a stylistic shortcut: if the transfer fails or requires extra restrictions, the claimed 0.296 bound and the improvement over S(c) < (7/2) log(1/c) would not follow. The authors should either prove the required hypotheses for beta_n, explicitly handling the coprimality structure relative to the moduli, or cite a theorem that establishes them for z-rough numbers.
  2. [Section 3.1, equation (3.2)] The parenthetical '(We may assume that Theorem 1.1 holds and that L(theta, nu) can take the value L(nu)-theta)' before (3.2) is ambiguous and appears to invoke the theorem being proved in order to delete conditions. If this is meant as a forward reference to the later cases, the text should say so explicitly and explain how the union of Cases 1-5 covers all admissible nu without assuming the conclusion. If Theorem 1.2 genuinely uses Theorem 1.1 as an input, then the proof structure must be reorganized to state clearly which result is established first. As written, a reader cannot determine whether (3.2) is a definition, a conditional claim, or a conclusion.
  3. [Section 4, numerical constant in Theorem 1.3] The proof of Theorem 1.3 ends with 'By numerical calculation, we have C(c, delta1) - 2 nu > 0.296', but no details of the calculation are given. The function t1(eta) is introduced only through the caption of Figure 1, and the subtracted term -0.233755 (log(1/(2 delta1)))^2 in the definition of C(c, delta1) is not derived. The constant 0.296 is the content of the theorem, so the numerical step is load-bearing. The authors should supply the explicit evaluation, a precise definition of t1(eta), and either the code or a certified interval computation for the integral; otherwise Theorem 1.3 is not checkable.
minor comments (5)
  1. [Title and abstract] The title and abstract contain typographical errors: 'WELL-F ACTORABLE', 'satisfties', and 'obatined' should be corrected.
  2. [Sections 3.1, 3.3, 3.5, 4] There are spelling errors in the body: 'suffciently' (Section 3), 'Follwing' (Section 3.3), 'accpetable' (Section 3.5), 'pragh' (Section 4), and 'Cauchy's inequity' (Sections 3.1 and 3.5) should be 'Cauchy's inequality'.
  3. [Theorem 1.4] The sentence 'where L(·) is the L(·) in Theorem 1.1' after the display of S(c) does not correspond to the displayed formula, which is fully explicit; this sentence should be deleted or clarified.
  4. [Section 5] In the derivation of Theorem 1.4, the notation A in M_i(A,P,d) and the role of the partition step sizes u_{i+1}-u_i are not defined locally; aligning the notation with [7] or adding definitions would improve readability.
  5. [References] The heading 'Acknowledegements' before the reference list is misspelled, and the placeholder text 'References' precedes the list; format both headings consistently.

Circularity Check

0 steps flagged · score 2.0 of 10

No construction-level circularity: the central distribution theorem is an application of external Pascadi and BFI inputs, while the flagged self-referential wording is an ambiguous forward reference and the unverified assumption transfers are correctness gaps, not circular reductions.

full rationale

The paper's central result, Theorem 1.1, is an amalgamation of estimates imported from Pascadi's triply-well-factorable convolution estimate [19, Proposition 4.4], his incomplete-Kloosterman-sum bound [20, Corollary 5.14], Wang's well-factorable convolution estimate [28, Proposition 3.2], and the BFI framework [2]. None of these is fitted to the target result and none is a renamed version of Theorem 1.1. The applications in Theorems 1.3 and 1.4 use Theorem 1.1 and Theorem 1.2 legitimately as inputs; the constants 0.296, 0.7404, and 0.8602 arise from numerical optimization of proven inequalities, not from fitting parameters to the claimed output. The phrase in Section 3.1, 'We may assume that Theorem 1.1 holds and that L(θ, ν) can take the value L(ν)−θ', is the closest thing to a self-referential step. Read literally it appears to assume the theorem being proved while forming (3.2). However, the final assembly in Section 3.6 states that for θ=0 'Theorem 1.1 comes from (3.2) (only see L1(0,ν))', meaning the θ=0 case uses only the L1(0,ν) component, while the L(ν)−θ term is a forward reference used after Theorem 1.1 is established for the θ>0 case of Theorem 1.2. Thus this is an ambiguous and poorly signposted proof organization, not a circular dependence in the actual chain. Similarly, the assertion in Section 3.2 that 'all assumptions (A_i) in [2] are satisfied whenever needed', and the silent Siegel–Walfisz check for the sequence 1_{l1∈L1}1_{P^-(l1)>z} in the application of Lemma 3.1, are unproved hypothesis-transfers from external work. If any hidden condition fails, the exponents L(ν) and L'(ν), and with them both applications, would not follow. That is a genuine correctness risk and a load-bearing gap, but it is not a reduction of a prediction to its own inputs or a fitted parameter renamed as a result, so it does not constitute circularity under the stated criteria.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central claims rest on Pascadi's quoted large sieve and Kloosterman sum estimates, on the BFI assumptions being inherited, and on unshown numerical evaluations; no new entities are invented.

free parameters (3)
  • c = 0.1348
    Choice of split parameter in the final numerical bound C(c,delta1) of Section 4; the claimed 0.296 bound depends on this particular value.
  • delta1 = 0.417
    Choice of lower cutoff in the integral for the final C(c,delta1) in Theorem 1.3; selected by numerical search.
  • nu (tolerance) = 0.000001
    Small tolerance parameter in Section 4 used to control the partition limits in the reduction to numerical integration.
assumptions (5)
  • domain assumption Assumption 2.3 (Exceptional Large Sieve): for tuples (q,N,Z,(a_n),A,Y), the eigenvalue-weighted Maass form large sieve bound holds.
    Quoted from Pascadi [20] and used in Lemmas 2.4-2.6 and throughout the proof of Theorem 1.1; the paper does not prove it.
  • domain assumption Lemma 2.6 (incomplete Kloosterman sums estimate) is valid as quoted.
    Quoted from Pascadi [20, Corollary 5.14] and used for the main estimates in Cases 2-5.
  • domain assumption Lemma 3.1 (triply-well-factorable convolution estimate) is valid as quoted.
    Quoted from Pascadi [19, Proposition 4.4] and used to prove the Case 1 parts of Theorems 1.1 and 1.2.
  • ad hoc to paper All assumptions (A_i) in Bombieri-Friedlander-Iwaniec [2] hold for the convolution weights in this paper.
    Stated in Section 3.2 with no verification; the BFI framework is the backbone of Cases 2-5.
  • standard math Hildebrand's estimate for Psi(x,y) (Lemma 2.1) holds uniformly in the stated ranges.
    Standard result quoted as [12, Theorem 1].

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Pith. "Pith review of Convolution-type Bombieri-Vinogradov theorem with well-factorable Weights, and its applications." pith.science (2026). https://pith.science/paper/SNAY63KG

@misc{pith2026260813299,
  author       = {Pith},
  title        = {Pith review of: Convolution-type Bombieri-Vinogradov theorem with well-factorable Weights, and its applications},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SNAY63KG}},
  note         = {Machine review of arXiv:2608.13299}
}
abstract

In 1986, Bombieri, Friedlander and Iwaniec famously obtained that primes are equidistributed in arithmetic progressions to moduli up to $x^{4/7-\varepsilon}$, using well-factorable weights. In this paper, we apply Pascadi's triply-well-factorable convolution estimate and his estimation of incomplete Kloosterman sums to generalize this result to a convolutionform, which improves Wang's result under certain conditions. As for application, we consider the asymptotic density of $\#\{n\leq x:P^+(n)<P^+(n+1)\}$ and $\#\{p\leq x:P^+(p-1)\geq p^c\}$, where $P^+(n)$ denote the largest prime factor of $n$. We show that for $x\rightarrow\infty$, one has\begin{align*} \#\{n\leq x:P^+(n)<P^+(n+1)\}>0.296x \end{align*} and \begin{align*} &\mathop{\lim\sup}_{x\rightarrow\infty} \frac{1}{\pi(x)}\#\{p\leq x:P^+(p-1)\geq p^c\}\\&\leq S(c)=\left\{ \begin{aligned} & \int_0^{1-c}\frac{2}{(5/8-189u/200)(1-u)}\mathrm{d}u, \quad&& \frac{184}{189}\leq c<1,\\& S\left(\frac{184}{189}\right)+\frac{10}{3}\log\frac{184}{189c},\quad&& 0.7404<c\leq \frac{184}{189}, \end{aligned}\right. \end{align*} where the function $S(c)$ satisfties $S(c)<\frac{7}{2}\log\frac{1}{c} $ for $0.7404<c<1$. The first result improves a previous result $0.280$ by the author (2026). The second result constitutes an improvement upon that of Ding and Wang (2025), who obatined $\mathop{\lim\sup}_{x\rightarrow\infty} \frac{1}{\pi(x)}\#\{p\leq x:P^+(p-1)\geq p^c\}\leq \frac{7}{2}\log\frac{1}{c}$.

Figures

Figures reproduced from arXiv: 2608.13299 by the authors.

Figure 1
Figure 1. η → t(η) = t1(1 − η) Finally, we have X n<x P +(n)<P +(n+1) 1 ≥ (C(c, δ1) − 2ν)x + o(x), where ν = 0.000001, C(c, δ1) is defined by C(c, δ1) := log 1 1 − c − 2 Z c 0 Z η 0 1 L′ (t, η)(1 − η) dtdη + Z 1/2 δ1 ρ  1 t  1 t dt + Z 1/2 c 1 − 2η + 2t1(η) 2(1 − η)(1 − η + t1(η))dη − 0.233755 log 1 2δ1 !2 − 0.46751 Z 1/2 δ1 log (2 − 2t) t dt, with c = 0.1348, δ1 = 0.417. By numerical calculation, we have C(c, δ1) − 2ν > 0.… view at source ↗

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