{"id":"d973cd4c-ba7c-42c2-8e1d-54352ae7180b","arxiv_id":"2506.22667","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Using Hooley neutralisers, a large sieve for quadratic characters is improved for multiplicatively weighted sequences, with applications to hyperbolic-region character sums.","lead":"A new large sieve bound is proved for double sums of Jacobi symbols with a multiplicative weight, using Hooley neutralisers to insert a Brun sieve. The bound improves prior estimates in a wide range and is applied to character sums over hyperbolic regions needed for a companion counting problem.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The stated range in which Theorem 1.1 improves Elliott's bound is false: for M=N^4, α=1/2, the new first term is larger by ~√(log N), so the advertised improvement does not hold in the stated range.","rationale":"The reader's weakest_assumption concerned uniformity of the O(1) term and the sieve fundamental lemma in Lemma 2.2; I do not find that to be the decisive obstruction, since the O(1) in (1.4) is normally understood as uniform and the fundamental lemma condition is satisfied. The reader also flagged the Lemma 4.1 Corollary 1.2 misapplication, which is real but appears fixable by applying Theorem 1.1 directly in the first integral range. The most load-bearing issue I identify is different: the paper's central advertised range for beating Elliott's bound is incorrect. A direct comparison of the dominant first terms shows that within the range N^2 log N ≤ M^{1-ε}(log M)^{1-α}, Theorem 1.1's bound is asymptotically larger than Elliott's bound by a power of log N whenever M is a fixed polynomial power of N. This affects the motivation and the stated utility of the main theorem, and it should be corrected before the paper is accepted. The theorem's inequality itself may be correct, and the applications may still work through averaging, but the claimed improvement over Elliott in the stated range is not supported. I recommend conditional acceptance pending a corrected statement of the improvement range or a revised comparison.","tokens_in":43435,"tokens_out":62789,"duration_ms":605631,"concrete_test":"Take α=1/2, ε=0.1, M=N^4, and N=e^{10} (so log N=10). The condition N^2 log N ≤ M^{1-ε}(log M)^{1-α} holds. Compute the dominant terms: Theorem 1.1 gives approximately 0.5 N^{4.5}√(log N) from its first term, while Elliott's bound (1.1) gives approximately N^{4.5}. The ratio is 0.5√(log N) > 1 and grows without bound as N grows. Repeating this comparison for any fixed α∈(0,1] and any fixed C>2 with M=N^C shows the ratio (log N)/(C log N)^{1-α} → ∞, confirming that the stated improvement range is false.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Theorem 1.1's first term is M N^{1/2}(log N)/(log M)^{1-α}. Elliott's bound (1.1) in the same region has dominant term M N^{1/2}. In the introduction's claimed improvement range, N^2 log N ≤ M^{1-ε}(log M)^{1-α}, we have log M ≍ log N; specifically, for M=N^4, log M = 4 log N. The ratio of the new first term to Elliott's first term is log N/(log M)^{1-α} = (log N)^α / 4^{1-α}, which tends to infinity for every fixed α>0. Since in this range the first term dominates both bounds (for C>2 the Elliott second term is lower order, and the Theorem 1.1 second term is also lower order), the theorem's bound is asymptotically worse than Elliott's, not better. The improvement over Elliott only occurs when log M ≥ (log N)^{1/(1-α)} (up to constants), i.e. M ≥ exp((log N)^{1/(1-α)}), which is far outside the stated polynomial range. This does not disprove the inequality in Theorem 1.1, but it invalidates the central advertised consequence that Theorem 1.1 improves Elliott's large sieve in the stated range. The inequality itself may still be useful in integral averages, but the headline improvement claim is demonstrably false for M=N^4, α=1/2.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims a new large-sieve bound for double character sums with one factor carrying a multiplicative weight f satisfying a prescribed average (1.4). The proof uses a Hooley-neutraliser pointwise inequality with Brun sieve coefficients, followed by the usual Cauchy--Schwarz and Pólya--Vinogradov route. The result is then applied to averages of L(1, chi . (./m)), to a general lemma on such averages, and to several character sums over hyperbolic regions that are needed in a companion paper on local solubility of diagonal quadrics. The paper is largely self-contained; the main theorem is proved from standard sieve tools, and the author's previous work is used only to frame the applications, so no circularity is apparent.","tokens_in":43752,"tokens_out":26218,"duration_ms":262145,"significance":"If the main theorem and its applications were correct as stated, the paper would supply a genuinely new weighted large sieve for quadratic characters and nontrivial estimates for character sums in hyper-skewed regions. The proof idea, combining Hooley neutralisers with the fundamental lemma of sieve theory, is coherent and not dependent on fitted constants. However, the central advertised improvement over Elliott's bound is incorrect for the stated range, and several applications invoke stated lemmas in parameter ranges where their hypotheses fail. The underlying inequality may still be useful in integral averages, but the paper's claims need substantial correction and re-verification before the applications can be accepted.","major_comments":[{"comment":"The claim that in the range N^2 log N ≤ M^{1-ε}(log M)^{1-α} Theorem 1.1 improves upon all of (1.1)–(1.3) is not supported. Take M = N^4 and α = 1/2; then log M = 4 log N, and the ratio of the new first term M N^{1/2}(log N)/(log M)^{1-α} to Elliott's dominant term M N^{1/2} equals (log N)^α / 4^{1-α}, which tends to infinity. In this range the first term dominates both the new second term and Elliott's second term, so the theorem's bound is asymptotically larger than Elliott's, not smaller. The improvement over (1.1) only begins when log M ≳ (log N)^{1/(1-α)} (up to constants), which is not satisfied by any polynomial M = N^C. This does not disprove the inequality in Theorem 1.1, but it invalidates the advertised central consequence; the paper should either prove a genuinely better pointwise bound or reframe the improvement as one for integral averages.","section":"§1.1 (paragraph after Theorem 1.1)"},{"comment":"The application of Corollary 1.2 with 'N = X and M = t' is invalid because t ≤ X^{1/2} implies M = t < X = N, while Corollary 1.2 requires M ≥ N. The displayed integrand matches Theorem 1.1 applied with the variables reversed (N = t, M = X), not Corollary 1.2. Moreover the second displayed term X^{1/3} t^{-1/2}(log t)^{1/2}/(log X)^{(1-α)/2} would require M^{1/2+ε} = X^{1/3} with ε = 1/6, i.e. ε = -1/6, so the bound as written does not follow from any stated lemma. Since Corollary 1.3 and the deductions in §4.2 depend on Lemma 4.1, this step must be corrected and the resulting bound recomputed.","section":"§4.1, proof of Lemma 4.1 around Eq. (4.2)"},{"comment":"The text says 'we apply Corollary 3.2 with ε = 1/10'; no Corollary 3.2 exists, and if Lemma 3.2 is intended, its hypothesis M ≥ N ≥ W ≥ 2 is violated. In T1 the m-range has M = (log Z)^{C2}, while the n1-range has N = n0 c0 with n0 ≥ Z^{10}, so M is a power of log log X and N is at least (log X)^{10 C1}; thus M ≪ N for large X. The displayed bound for T1 does not follow from Lemma 3.2 and appears to be an unsupported new estimate. The same problem occurs in T2. Since Corollary 1.4 is used in Proposition 5.13, this gap propagates into the main applications of §5 and §6.","section":"§4.2, proof of Corollary 1.4 (estimates for T1 and T2)"},{"comment":"The claimed uniformity 'the implied constant depends at most on ε' is stronger than what is proved. Lemma 2.2 uses the fundamental lemma with the bound ∏_{w<p≤z}(1-f(p)/p)^{-1} ≪ (log z / log w)^α, whose implicit constant is exp(2C) where C is the O(1) constant in (1.4). If the O(1) in (1.4) is allowed to depend on f, then the final constant depends on that O(1) and on α, not only on ε. The same dependence is inherited by Theorem 1.1 and Corollary 1.2. For the applications with f = 1/τ this can be made explicit, but the theorem as stated needs either a uniformity assumption on the O(1) constant or a modified conclusion.","section":"§2, Lemma 2.2 and Theorem 1.1"}],"minor_comments":[{"comment":"Reference [25] appears to be missing the author's name: the entry 'Spécialisation des éléments de Br2(Q(T1,...,Tn))' is a paper by Serre, not by Selmer, and the bibliography entry should be completed and separated from [24].","section":"References"},{"comment":"In the treatment of H2(X), the text says 'Lemma 3.3 for the sums over n2 and n4'; there is no n4 in (5.4)–(5.6), and the second sum should be over n3.","section":"§5.2, proof of Lemma 5.8"},{"comment":"The condition '1 ≤ c0, c1 ≤ (log X)^{C1}/32' is likely a typo; the proof and Proposition 5.13 suggest the intended bound is c0, c1 ≤ (log X)^{C1/32}.","section":"Statement of Corollary 1.4"},{"comment":"The notation ∑_{χ mod 8} χ(q0) ~L_{r0}(1, χ02 χ) uses the same symbol χ both as an outer summation index and inside the character product; renaming the outer character (for example, χ_8) would remove an unnecessary ambiguity.","section":"§6.3, Lemma 6.4 and Corollary 6.5"}],"recommendation":"major_revision","confidential_remarks":"The paper contains a genuinely interesting new weighted large-sieve inequality, but the introduction's headline comparison with Elliott's bound is false as stated, and several key applications cite lemmas outside their valid parameter ranges. The issues appear fixable in a revision: the main theorem may still hold, and the applications can likely be repaired by tracking variable roles and epsilon exponents carefully. I would not reject outright, but I would require a thorough revision before further consideration."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Thanks for the report. I read the paper closely and here is my take.\n\nThe main result, Theorem 1.1, is a real new bound: a large sieve for quadratic characters with a multiplicative weight f(m) and a parameter α, proved via Hooley neutralisers. The proof is coherent, and Lemma 2.1 is a clean way to insert the Brun sieve. The paper is honest in that the inequality itself is not fitted or circular; the applications to hyperbolic character sums are extensive and tailored to the author's programme.\n\nBut the central advertised consequence is wrong. The introduction claims Theorem 1.1 improves Elliott's bound in the range N^2 log N ≤ M^{1-ε} (log M)^{1-α}. For M=N^4 and α=1/2, the new first term is M N^{1/2} (log N)/(4 log N)^{1/2} = M N^{1/2} (log N)^{1/2}/2, which is larger than Elliott's first term M N^{1/2} by a factor ~(log N)^{1/2}. So the theorem is asymptotically worse, not better, in the stated range. The improvement only kicks in when log M is a power of log N, e.g., M ≥ exp((log N)^{1/(1-α)}). This does not disprove the inequality, but it invalidates the headline claim.\n\nThe reader also found a misapplication in Lemma 4.1: it says it uses Corollary 1.2 with N=X and M=t, which violates M ≥ N. The displayed bound matches Theorem 1.1 with N=t and M=X. I agree that is a typo, but it should be fixed. While checking, I also could not reproduce the X^{1/3} coefficient in the second term of the displayed integral from Theorem 1.1 as stated; it looks like a typo for X^{2/3} or comes from a different application. The second term is lower-order, so this is minor, but a referee should verify.\n\nThe dependence of the implied constant in Theorem 1.1 on the O(1) in (1.4) is also glossed over; the proof inherits that constant, so \"depends only on ε\" is stronger than demonstrated unless the O(1) is absolute.\n\nThe paper is a solid technical engine for the author's programme, and the neutraliser technique deserves a serious referee. But the claims need major revision before the paper can be trusted at face value. I would send it to peer review, but with a firm request to correct the range claim and the Lemma 4.1 details.","headline":"A genuinely new neutraliser-large sieve, but the advertised improvement range over Elliott is false and the paper needs corrections before its claims can be trusted.","tokens_in":44278,"tokens_out":8586,"would_cite":false,"duration_ms":74670,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11N36","11A25","11L40"],"pacs":[],"model":"deepseek-v4-flash","headline":"Combining Hooley neutralisers with the large sieve for quadratic characters gives a weighted character-sum bound that beats the classical estimates in hyper-skewed ranges, and the paper uses it to control character sums over hyperbolic…","keywords":["quadratic large sieve","quadratic characters","Hooley neutralisers","Brun sieve","multiplicative functions","character sums","hyperbolic regions","local solubility"],"falsifier":"Directly compute the double sum for a concrete $f$ with known $\\alpha$, say $f=1/\\tau$ ($\\alpha=1/2$), in ranges with $N^2\\log N\\le M^{1-\\epsilon}(\\log M)^{1/2}$; if the empirical maximum exceeds $MN^{1/2}(\\log N)/(\\log M)^{1/2}$ by a positive power of $\\log M$, the claimed saving is false. Alternatively, check numerically whether $\\sum_{p\\le X}f(p)/p-\\alpha\\log\\log X$ stays bounded as $X$ grows for the chosen $f$, since Lemma 2.2 depends on that uniformity.","tokens_in":43217,"feed_emoji":"🧮","tokens_out":9912,"duration_ms":94024,"temperature":0.7,"pith_summary":"The paper claims that the large sieve for quadratic characters can be sharpened when the $m$-side sequence carries a multiplicative weight $f(m)$ whose average over primes is $\\alpha\\log\\log X+O(1)$. Theorem 1.1 proves a bound whose first term carries a saving of $(\\log M)^{1-\\alpha}$, and in the hyper-skewed range $N^2\\log N \\le M^{1-\\epsilon}(\\log M)^{1-\\alpha}$ this beats all three classical bounds (1.1)--(1.3). The mechanism is to insert Brun sieve coefficients through a pointwise Hooley-neutraliser inequality, so the saving comes from the multiplicative structure of $f$ rather than from new information about character sums alone. The paper then uses the improved sieve to control error terms in character sums over hyperbolic regions, which are the input to the companion asymptotic $N(B)\\sim cB^2\\log\\log B/\\log B$ for the local solubility problem (1.10).","feed_headline":"Weighted large sieve beats older bounds when N^2≤M.","feed_subtitle":"Inserting a Brun sieve into the character-sum second moment saves a power of log M.","key_machinery":"The load-bearing object is the pointwise Hooley neutraliser inequality $f(n)\\le \\sum_{d\\mid n,\\,d\\mid P(z)}\\lambda_d^+\\hat f(d)$, where $\\hat f(d)=\\prod_{p\\mid d}(1-f(p))$ and $(\\lambda_d^+)$ are the Brun upper-bound sieve coefficients supported on $[1,y]$ with $y=z^{10}$. Because the inequality is pointwise, the proof can insert the sieve directly into the second moment of the character sum; then Lemma 2.2 evaluates the diagonal contribution by the fundamental lemma of sieve theory, turning the average of $\\hat f$ into the factor $(\\log M)^{-(1-\\alpha)}$. Shiu's theorem supplies the companion estimate $\\sum_{m\\le M}f(m)\\ll M/(\\log M)^{1-\\alpha}$. This mechanism, rather than a better treatment of the off-diagonal characters, is what creates the saving.","core_discovery":"On the paper's own terms, the central discovery is that a quadratic character double sum improves when the $m$-variable is weighted by a multiplicative $f$ satisfying $0\\le f(p)\\le 1$, $f(p^m)\\le f(p)$, and $\\sum_{p\\le X} f(p)/p = \\alpha\\log\\log X+O(1)$. The bound is $\\sum_{n\\le N}\\sum_{m\\le M} a_n b_m f(m)(n/m) \\ll_\\epsilon MN^{1/2}(\\log N)/(\\log M)^{1-\\alpha} + M^{1/2+\\epsilon}N^{3/2}(\\log N)^{1/2}/(\\log M)^{(1-\\alpha)/2}$, with the first term dominant exactly when $N^2\\log N \\le M^{1-\\epsilon}(\\log M)^{1-\\alpha}$. In that range the result improves on the three classical large-sieve bounds (1.1)--(1.3), and the harmonic version in Corollary 1.2 carries the same saving together with convergence from the factor $1/n$. The paper derives from it estimates such as $\\sum_{1<m\\le X}\\mu^2(2m)\\tau(m)^{-1}L(1,(\\cdot/m))\\ll X/\\sqrt{\\log X}$, which it reads as independence between $\\tau(m)$ and the $L$-value.","pith_inferences":["The neutraliser insertion is not specific to Jacobi symbols: the same second-moment argument should yield a $(\\log M)^{-(1-\\alpha)}$ saving for any family of real characters where the diagonal sum can be evaluated by a sieve.","One reading of the bound is that the effective mass of a multiplicative $f$ with average $\\alpha$ behaves like a set of density $(\\log M)^{-(1-\\alpha)}$, so the saving is equivalent to shrinking $M$ by that factor before applying the ordinary large sieve.","A numerical check with $f=1/\\tau$ ($\\alpha=1/2$) in the range $N^2\\log N\\le M^{1-\\epsilon}(\\log M)^{1/2}$ would test whether the first-term saving is visible in small ranges or only asymptotically."],"forward_implications":["In the range $N^2\\log N\\le M^{1-\\epsilon}(\\log M)^{1-\\alpha}$, Theorem 1.1 improves on the three classical quadratic-character large-sieve bounds (1.1)--(1.3).","Corollary 1.2 encodes both the multiplicative saving and the convergence of $\\sum 1/n\\,(n/m)$, making it usable in hyper-skewed regions where the classical bounds fail.","Corollary 1.3 bounds the average of $\\tau(m)^{-1}L(1,(\\cdot/m))$ by $X/\\sqrt{\\log X}$, matching what independence of the two factors would predict.","The error terms in Propositions 5.7, 5.9, 5.13 and 6.3, 6.6 are strong enough to feed into the companion proof of $N(B)\\sim cB^2\\log\\log B/\\log B$ for the local solubility count (1.10)."],"supporting_citations":[{"why":"supplies the bound (1.1) that Theorem 1.1 improves in the skewed range.","marker":"[8, 14]"},{"why":"supplies the squarefree mean-value bound (1.2) used as a complement.","marker":"[14]"},{"why":"supplies the bound (1.3) used as a baseline and in the applications.","marker":"[11]"},{"why":"provides the Hooley neutraliser proposition that Lemma 2.1 modifies.","marker":"[27]"},{"why":"gives the bound $\\sum_{m\\le M} f(m)\\ll M/(\\log M)^{1-\\alpha}$ used in the proof of Theorem 1.1.","marker":"[26]"},{"why":"supplies the fundamental lemma of sieve theory used in Lemma 2.2.","marker":"[17]"},{"why":"supplies the Selberg--Delange type character-sum lemma used for small conductors.","marker":"[20]"}],"fun_headline_variants":["Weighted quadratic large sieve saves log power when N² ≤ M","Hooley neutralisers boost quadratic character large sieve","Improved bound for quadratic character double sums when M ≥ N²","Sieve weighting yields quadratic character sum saving for M large","Neutralisers and weighted sieve beat classical large-sieve bounds"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof needs the $O(1)$ term in the prime-average condition $\\sum_{p\\le X}f(p)/p=\\alpha\\log\\log X+O(1)$ to be uniform in $X$, and it applies the fundamental lemma of sieve theory to products over primes up to $X^{\\epsilon/10}$ using only that average; if uniformity fails, the stated $(\\log M)^{-(1-\\alpha)}$ saving in the first term is not justified.","fun_headline_variants_meta":{"raw":{"variants":["Weighted quadratic large sieve saves log power when N² ≤ M","Hooley neutralisers boost quadratic character large sieve","Improved bound for quadratic character double sums when M ≥ N²","Sieve weighting yields quadratic character sum saving for M large","Neutralisers and weighted sieve beat classical large-sieve bounds"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000608,"raw_usage":{"total_tokens":2776,"prompt_tokens":836,"completion_tokens":1940,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":452,"completion_tokens_details":{"reasoning_tokens":1857}},"tokens_in":452,"tokens_out":1940,"duration_ms":14282,"temperature":1.0,"reasoning_tokens":1857,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T22:03:03.048127+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Directly compute the double sum for a concrete $f$ with known $\\alpha$, say $f=1/\\tau$ ($\\alpha=1/2$), in ranges with $N^2\\log N\\le M^{1-\\epsilon}(\\log M)^{1/2}$; if the empirical maximum exceeds $MN^{1/2}(\\log N)/(\\log M)^{1/2}$ by a positive power of $\\log M$, the claimed saving is false. Alternatively, check numerically whether $\\sum_{p\\le X}f(p)/p-\\alpha\\log\\log X$ stays bounded as $X$ grows for the chosen $f$, since Lemma 2.2 depends on that uniformity.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the squarefree mean-value bound (1.2) used as a complement."},{"cited_title":"Friedlander, H","cited_arxiv_id":null,"evidence_quote":"supplies the bound (1.3) used as a baseline and in the applications."},{"cited_title":"Sofos,Serre’s problem on the density of isotropic fibres in conic bundles, Proc","cited_arxiv_id":null,"evidence_quote":"provides the Hooley neutraliser proposition that Lemma 2.1 modifies."},{"cited_title":"Shiu,A Brun-Titchmarsh theorem for multiplicative functions,J","cited_arxiv_id":null,"evidence_quote":"gives the bound $\\sum_{m\\le M} f(m)\\ll M/(\\log M)^{1-\\alpha}$ used in the proof of Theorem 1.1."},{"cited_title":"Iwaniec, E","cited_arxiv_id":null,"evidence_quote":"supplies the fundamental lemma of sieve theory used in Lemma 2.2."},{"cited_title":"The leading constant for rational points in families","cited_arxiv_id":"2210.13559","evidence_quote":"supplies the Selberg--Delange type character-sum lemma used for small conductors."}],"review_version":1}