{"id":"40cc6710-b9de-4413-ba7f-86b51886b338","arxiv_id":"2608.13299","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A new convolution-type Bombieri-Vinogradov theorem with well-factorable weights improves the lower density for P^+(n)<P^+(n+1) from 0.280 to 0.296 and the upper density for P^+(p-1)>=p^c from (7/2)log(1/c) to a smaller S(c).","lead":"This paper proves a convolution-type Bombieri-Vinogradov theorem with well-factorable weights, reaching an exponent of distribution at least 4/7, and uses it to improve two constants in problems about largest prime factors of neighbouring integers and shifted primes. The main applications are a new lower density 0.296 for n with P^+(n) < P^+(n+1) and a smaller limsup bound for primes p with P^+(p-1) >= p^c.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The distribution theorem's core is an unverified transfer of BFI [2] hypotheses to the rough-number sequence β_n=1_{P^-(n)>z}; if that transfer fails, the exponents L(ν) and both applications collapse, so this gap is load-bearing.","rationale":"The reader's weakest_assumption already identifies the same central risk: the paper assumes that the full technical apparatus of Bombieri–Friedlander–Iwaniec [2] carries over to the new convolution weights, with only the sentence \"all assumptions (A_i) in [2] are satisfied whenever needed\" as support. I selected this as the single load-bearing concern because it sits at the root of the logical chain: (3.2), (3.5), (3.7), (3.11), and (3.12) all feed into L(ν), which then feeds L(θ,ν), L'(ν), and finally Theorems 1.3 and 1.4. A numerical inaccuracy in the evaluation of C(c,δ1) would only weaken Theorem 1.3 and is checkable by recomputing the displayed integrals; a failure of the BFI assumption would invalidate the distribution apparatus itself. The paper gives no proof that rough numbers satisfy the Siegel–Walfisz condition, and this cannot be taken from [2] since [2] treats primes. The \"We may assume Theorem 1.1 holds\" aside near (3.2) is not, on my reading, a logically circular step, because Section 3.6 presents the cases as a maximum of independently established ranges; still, the exposition should disambiguate it. Since the gap is real but potentially repairable by a lemma verifying the needed assumptions for z-rough sequences, the conditional verdict remains appropriate.","tokens_in":24009,"tokens_out":13095,"duration_ms":129034,"concrete_test":"Extract the explicit hypotheses (A_i) from [2, §1–2] and the Siegel–Walfisz condition from [19, Prop. 4.4], then prove for α_n=1_{n∼L1, P^-(n)>z} the uniform bound ∑_{q≤(log x)^B} max_{(a,q)=1} |∑_{n≤N, n≡a mod q, P^-(n)>z} 1 − φ(q)^{-1}∑_{n≤N, (n,q)=1, P^-(n)>z} 1| ≪_A N (log N)^{-A} for every A,B. If this bound fails in some range N=x^ν, identify the range and recompute L(ν) in (3.5), (3.7), (3.11), and (3.12) with the weaker error term; if no proof exists, replace β_n by a sequence satisfying the axiom and recompute the exponents.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 3.2 states \"we note that all assumptions (A_i) in [2] are satisfied whenever needed\" immediately after replacing the BFI sequence with β_n = 1_{l1∈L1}1_{P^-(l1)>z}. No derivation is given. The BFI proof relies on a system of hypotheses on the input sequence (Siegel–Walfisz, Type I/II bounds, and exponential-sum conditions), and the new rough-number indicator is not the prime sequence for which those hypotheses were originally established. The same omission occurs in the application of Lemma 3.1, where α_n is set to 1_{l1∈L1}1_{P^-(l1)>z} and the lemma's Siegel–Walfisz condition is silently assumed. Theorems 1.3 and 1.4 inherit every decrease in L(ν) or L'(ν), so this is not a stylistic shortcut: a missing condition could lower the level and undo the claimed 0.296 bound and the improvement over S(c)<(7/2)log(1/c). The phrase \"We may assume that Theorem 1.1 holds\" near (3.2) is also ambiguous and should be clarified as a forward reference to the later cases, not as a circular dependence.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":24305,"tokens_out":8392,"duration_ms":80444,"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":[{"comment":"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.","section":"Section 3.2 (and Section 3.1, application of Lemma 3.1)"},{"comment":"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.","section":"Section 3.1, equation (3.2)"},{"comment":"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.","section":"Section 4, numerical constant in Theorem 1.3"}],"minor_comments":[{"comment":"The title and abstract contain typographical errors: 'WELL-F ACTORABLE', 'satisfties', and 'obatined' should be corrected.","section":"Title and abstract"},{"comment":"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'.","section":"Sections 3.1, 3.3, 3.5, 4"},{"comment":"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.","section":"Theorem 1.4"},{"comment":"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.","section":"Section 5"},{"comment":"The heading 'Acknowledegements' before the reference list is misspelled, and the placeholder text 'References' precedes the list; format both headings consistently.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper is squarely within the scope of math.NT and the claimed results would be of genuine interest. My recommendation of major_revision is driven by the two load-bearing gaps described above: the unverified transfer of BFI's hypotheses to the rough-number sequence, and the opaque numerical validation of the 0.296 constant. Both appear repairable within the manuscript's scope. I would also ask the editor to have the authors clarify the overlap between the present work, the author's preprint [30], and Pascadi's results, so that the incremental novelty is transparent."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First, the one-sentence summary: this paper does what it claims. Theorem 1.1 is a genuine convolution-type analogue of Bombieri–Friedlander–Iwaniec, with a piecewise level L(ν) that is at least 4/7 for all ν and larger on several intervals, and Theorem 1.2 extends this to allow a divisor variable D=x^θ. The two applications are real improvements, not cosmetic: Theorem 1.3 raises the lower density for P+(n)<P+(n+1) from the author's own 0.280 to 0.296, and Theorem 1.4 replaces Ding–Wang's (7/2)log(1/c) bound with a smaller piecewise S(c). The improvements are modest, but they are record constants, and they follow from the distribution theorem rather than being fitted to it.\n\nThe architecture follows BFI and Pascadi closely, which is the right machinery. The paper uses Pascadi's triply-well-factorable estimates and exceptional large sieve deliberately, and the citation pattern looks solid. Self-citing the author's earlier paper [30] to improve its constant is legitimate, not a red flag. There is no sign of parameter-fitting in the main theorems.\n\nThe soft spot is exactly where the stress-test put it. In Section 3.2, after setting β_n = 1_{l1∈L1} 1_{P^-(l1)>z}, the paper says that all assumptions (A_i) in [2] are satisfied whenever needed. That is load-bearing and it is asserted, not verified. The BFI hypotheses were originally designed for the prime sequence; the rough-number indicator is a different object, and Lemma 3.1 silently assumes a Siegel–Walfisz condition for the same kind of sequence. This may well be true, and it may follow from known results, but the paper needs to say how. If any condition fails, the level L(ν) drops and both applications are undone. The parenthetical in Section 3.1, 'We may assume that Theorem 1.1 holds,' also reads as circular if taken literally. It is most plausibly a forward reference to the later cases, but in a proof that kind of ambiguity should be cleaned up. Section 3.6's derivation of L(θ,ν)=L(ν)−θ for θ≥2/7−ε is likewise a sketch rather than a full argument.\n\nSeparately, the numerical claims are not documented: the constant C(0.1348, 0.417)>0.296 and the values 0.7404 and 0.8602 are asserted with no code and no displayed arithmetic. That is minor in this field, but a referee should be able to reproduce them.\n\nWho is this for: people working on primes in arithmetic progressions to large moduli, smooth-weighted distribution theorems, and Erdős–Turán/Pomerance-type questions. It deserves a serious referee. If the hypothesis-transfer in Section 3.2 can be justified, the paper stands; if not, the record constants are at risk. Send it to peer review, and ask the referee to focus on that transfer and on the numerical verification.","headline":"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.","tokens_in":24869,"tokens_out":3143,"would_cite":true,"duration_ms":35346,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11N13","11N25","11N36","11N37"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["primes in arithmetic progressions","Bombieri–Vinogradov theorem","well-factorable weights","linear sieve","largest prime factor","consecutive integers","shifted primes","Dickman function"],"falsifier":"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.","tokens_in":23783,"feed_emoji":"🔢","tokens_out":12521,"duration_ms":122546,"temperature":0.7,"pith_summary":"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 $P^+(n)<P^+(n+1)$ (and the same for the reverse inequality), and the shifted primes $p\\le x$ with $P^+(p-1)\\ge p^c$ have upper density at most $S(c)<\\frac72\\log\\frac1c$ for every $c>0.7404$. 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.","feed_headline":"Prime distribution result lifts n vs n+1 record to 0.296","feed_subtitle":"Smooth-weighted primes stay equidistributed to moduli near x^{4/7}; two density records fall.","key_machinery":"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.","core_discovery":"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 $c>0.7404$.","pith_inferences":["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 $S(c)$."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the classical 4/7 well-factorable method whose technical assumptions Section 3.2 re-uses and whose Cases 2–5 are adapted with new averaging lemmas.","marker":"[2]"},{"why":"Supplies the triply-well-factorable convolution estimate (Lemma 3.1) and the factorization of sieve weights used to reach level 3/5.","marker":"[19]"},{"why":"Supplies the exceptional large sieve Assumption 2.3, the truncated Poisson lemma, and the incomplete Kloosterman sum bounds used in Lemmas 3.2–3.4.","marker":"[20]"},{"why":"Provides the previous convolution-type estimate at exponent 4/7 used on two intervals of $\\nu$, and the earlier consecutive-integer density method.","marker":"[28]"},{"why":"Provides the $S_A-S_B-S_C$ decomposition and numerical framework from which the 0.296 lower bound is derived.","marker":"[30]"},{"why":"Sets up the shifted-prime upper-bound method and the baseline $S(c)\\le\\frac72\\log\\frac1c$ that Theorem 1.4 improves.","marker":"[7]"},{"why":"Supplies the triply-well-factorable framework and the comparison proposition that Lemma 3.1 refines.","marker":"[17]"},{"why":"Supplies the Dickman–de Bruijn estimate used to discard the small-prime-factor part of $l$.","marker":"[12]"},{"why":"Defines the upper-bound well-factorable linear sieve weights and their variant used for $L'(\\nu)$ and for the applications.","marker":"[13]"}],"fun_headline_variants":["Density of n with P^+(n)<P^+(n+1) exceeds 0.296x","New bound: #{n≤x: P^+(n)<P^+(n+1)} > 0.296x","Well-factorable weights lift prime divisor inequality to 0.296","Convolution sieve pushes density of P^+ inequality to 0.296","Prime distribution improvement: 0.296 density for n vs n+1"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Density of n with P^+(n)<P^+(n+1) exceeds 0.296x","New bound: #{n≤x: P^+(n)<P^+(n+1)} > 0.296x","Well-factorable weights lift prime divisor inequality to 0.296","Convolution sieve pushes density of P^+ inequality to 0.296","Prime distribution improvement: 0.296 density for n vs n+1"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001431,"raw_usage":{"total_tokens":5980,"prompt_tokens":1360,"completion_tokens":4620,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":976,"completion_tokens_details":{"reasoning_tokens":4502}},"tokens_in":976,"tokens_out":4620,"duration_ms":36194,"temperature":1.0,"reasoning_tokens":4502,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:18:48.217944+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Bombieri, J","cited_arxiv_id":null,"evidence_quote":"Supplies the classical 4/7 well-factorable method whose technical assumptions Section 3.2 re-uses and whose Cases 2–5 are adapted with new averaging lemmas."},{"cited_title":"Pascadi,Large sieve inequalities for exceptional Maass forms and the greatest prime factor of n2 + 1, Forum Math","cited_arxiv_id":null,"evidence_quote":"Supplies the exceptional large sieve Assumption 2.3, the truncated Poisson lemma, and the incomplete Kloosterman sum bounds used in Lemmas 3.2–3.4."},{"cited_title":"Wang,Sur les plus grands facteurs premiers d’entiers cons´ ecutifs, Mathematika,64(2018), no","cited_arxiv_id":null,"evidence_quote":"Provides the previous convolution-type estimate at exponent 4/7 used on two intervals of $\\nu$, and the earlier consecutive-integer density method."},{"cited_title":"An improvement on the largest prime factors of consecutive integers","cited_arxiv_id":"2607.16032","evidence_quote":"Provides the $S_A-S_B-S_C$ decomposition and numerical framework from which the 0.296 lower bound is derived."},{"cited_title":"Ding and Z","cited_arxiv_id":null,"evidence_quote":"Sets up the shifted-prime upper-bound method and the baseline $S(c)\\le\\frac72\\log\\frac1c$ that Theorem 1.4 improves."},{"cited_title":"Maynard,Primes in Arithmetic Progressions to Large Moduli II: Well-Factorable Estimates, Mem","cited_arxiv_id":null,"evidence_quote":"Supplies the triply-well-factorable framework and the comparison proposition that Lemma 3.1 refines."},{"cited_title":"Hildebrand,On the number of positive integers≤xand free of prime factors> y, J","cited_arxiv_id":null,"evidence_quote":"Supplies the Dickman–de Bruijn estimate used to discard the small-prime-factor part of $l$."},{"cited_title":"Iwaniec,A new form of the error term in the linear sieve, Acta Arith.,37(1980), 307–320","cited_arxiv_id":null,"evidence_quote":"Defines the upper-bound well-factorable linear sieve weights and their variant used for $L'(\\nu)$ and for the applications."}],"review_version":1}