Pith. sign in

REVIEW 4 major objections 4 minor 31 references

An improved large sieve for quadratic characters via Hooley neutralisers and its applications

T0 review · 4 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2506.22667 v2 pith:MLMTTWGJ submitted 2025-06-27 math.NT

Cameron Wilson This is my paper · ORCID
classification math.NT MSC 11N3611A2511L40
keywords quadraticlargesievecharactersHooleyneutralisersBrunmultiplicativefunctionscharactersumshyperbolicregionslocalsolubility
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 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).

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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).

Reading between the lines

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

  • 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.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

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.

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 (4)
  1. [§1.1 (paragraph after Theorem 1.1)] 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.
  2. [§4.1, proof of Lemma 4.1 around Eq. (4.2)] 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.
  3. [§4.2, proof of Corollary 1.4 (estimates for T1 and T2)] 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.
  4. [§2, Lemma 2.2 and Theorem 1.1] 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.
minor comments (4)
  1. [References] 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].
  2. [§5.2, proof of Lemma 5.8] 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.
  3. [Statement of Corollary 1.4] 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}.
  4. [§6.3, Lemma 6.4 and Corollary 6.5] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 1.1 is derived from external sieve results, Shiu's theorem and Polya-Vinogradov; alpha is an input parameter and the author's self-citations frame applications rather than support the main estimate.

full rationale

The main estimate (Theorem 1.1) is proved from Lemma 2.1 (a neutraliser inequality attributed to [27, Proposition 4.1], i.e. Sofos), Lemma 2.2 (fundamental lemma of sieve theory from [17]), Shiu's theorem [26], and Polya-Vinogradov. The parameter alpha is an arbitrary input satisfying the stated average (1.4); it is not fitted to the quantity being bounded. The saving (log M)^{-(1-alpha)} is produced by applying the same input average twice (once via Shiu, once via the sieve with hat f), but the conclusion concerns a genuinely different object - a double character sum with arbitrary bounded coefficients - and is not equivalent by construction to the input (1.4). The author's self-citations [30] and [31] are used to describe applications and the companion proof of Theorem 1.5; they are not load-bearing in the proof of Theorem 1.1. Concerns raised about non-uniformity of the O(1) constant in (1.4) and about the claim that Theorem 1.1 improves Elliott's bound in the stated polynomial range are correctness/validity issues, not circularity: they do not make any step reduce to its own input by definition.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

Theorem 1.1 rests on standard sieve tools, classical character sum estimates, and the stated mean-value hypothesis. No fitted constants or new entities are introduced; alpha is an input parameter of the multiplicative function.

assumptions (6)
  • standard math Shiu's theorem: sum_{n ≤ X} f(n) ≪ (X / log X) exp(sum_{p ≤ X} f(p)/p) under the stated conditions on f.
    Invoked in equation (2.4) and used to bound the sum of f(m) and auxiliary divisor-function sums; cited to [26].
  • standard math Fundamental lemma of sieve theory, [17, Fundamental Lemma 6.3].
    Used in Lemma 2.2 to evaluate the weighted sum of Brun sieve coefficients over d; requires the product condition checked in the proof.
  • standard math Polya-Vinogradov bound for non-principal characters modulo q.
    Used in Section 2 to bound the non-square diagonal contribution in the proof of Theorem 1.1.
  • standard math Siegel-Walfisz style asymptotic for sums of chi(n)/tau(n), Lemma 3.3 from [20].
    Used throughout Sections 5 and 6; relies on a zero-free region and Siegel's theorem, hence the implied constants are ineffective.
  • domain assumption The multiplicative function f satisfies the mean value condition sum_{p ≤ X} f(p)/p = alpha log log X + O(1).
    This is the central hypothesis of Theorem 1.1; without a uniform O(1) constant, the saving (log M)^{-(1-alpha)} is not controlled.
  • domain assumption The inequalities 0 ≤ f(p) ≤ 1 and f(p^m) ≤ f(p) hold for all primes p and all m ≥ 1.
    Needed for Lemma 2.1 and for Shiu's theorem; excludes functions with large values at prime powers.

how reviews work

0 comments
Cite this review

Pith. "Pith review of An improved large sieve for quadratic characters via Hooley neutralisers and its applications." pith.science (2026). https://pith.science/paper/MLMTTWGJ

@misc{pith2026250622667,
  author       = {Pith},
  title        = {Pith review of: An improved large sieve for quadratic characters via Hooley neutralisers and its applications},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MLMTTWGJ}},
  note         = {Machine review of arXiv:2506.22667}
}
read the original abstract

We combine Hooley neutralisers and the large sieve for quadratic characters. We give applications to character sums with a hyperbolic height condition.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

31 extracted references · 30 canonical work pages

  1. [1]

    Bhargava,A positive proportion of plane cubics fail the Hasse Principle, arXiv, (2014),https://arxiv

    M. Bhargava,A positive proportion of plane cubics fail the Hasse Principle, arXiv, (2014),https://arxiv. org/abs/1402.1131

    arXiv 2014

  2. [2]

    T. D. Browning, D. R. Heath-Brown,Density of rational points on a quadric bundle inP3× P3, Duke Math. J. 169(16), pp.3099-3165, (2020)

    work page 2020

  3. [3]

    T. D. Browning, D. R. Heath-Brown,The geometric sieve for quadrics, Forum Math. 33(1), pp.147-165, (2021)

    work page 2021

  4. [4]

    T. D. Browning, D. Loughran, Sieving rational points on varieties, Trans. Amer. Math. Soc. 371(8), pp.5757–5785, (2019)

    work page 2019

  5. [5]

    T. D. Browning, J. Lyczak, R. Sarapin,Local solubility for a family of quadrics over a split quadric surface, Involve 16(2), pp.331-342, (2023),https://doi.org/10.2140/involve.2023.16.331

    work page doi:10.2140/involve.2023.16.331 2023

  6. [6]

    T. D. Browning, J. Lyczak, A. Smeets,Paucity of rational points on fibrations with multiple fibres, arXiv, (2023), https://arxiv.org/abs/2310.01135

    work page Pith review arXiv 2023

  7. [7]

    Local solubility in generalised Ch\^atelet varieties

    K. Destagnol, J. Lyczak, E. Sofos,Local solubility in generalised Châtelet varieties, arXiv, (2025), https: //arxiv.org/abs/2504.11388

    work page Pith review arXiv 2025

  8. [8]

    P. D. T. A. Elliott,On the mean value off (p). Proc. London Math. Soc.21(3), pp.28-96, (1970)

    work page 1970

Show all 31 references
  1. [9]

    Fouvry, J

    É. Fouvry, J. Klüners, On the 4-rank of class groups of quadratic number fields.Invent. Math. 167(3), pp.455-513, (2007)

    2007

  2. [10]

    Franke, Y.I

    J. Franke, Y.I. Manin and Y. Tschinkel,Rational points of bounded height on Fano varieties, Invent. Math. 95, pp.421–435, (1989)

    1989

  3. [11]

    Friedlander, H

    J. Friedlander, H. Iwaniec,Ternary quadratic forms with rational zeros.Journal de théorie des nombres de Bordeaux 22(1), pp.97-113, (2010)

    2010

  4. [12]

    C. R. Guo,On solvability of ternary quadratic forms, Proc. London Math. Soc.70(2), pp.241–263 (1995)

    1995

  5. [13]

    D. R. Heath-Brown,The size of Selmer groups for the congruent number problem, Invent. Math., 111(1) pp.171-196, (1993)

    1993

  6. [14]

    D. R. Heath-Brown,A mean value estimate for real character sums.Acta Arith. 72(3), pp.235-275, (1995)

    1995

  7. [15]

    Hooley,On ternary quadratic forms that represent zero, Glasgow Math

    C. Hooley,On ternary quadratic forms that represent zero, Glasgow Math. J.35(1), pp.13-23 (1993)

    1993

  8. [16]

    , On ternary quadratic forms that represent zero. II., J. reine angew. Math.602, pp.179-225, (2007)

    2007

  9. [17]

    Iwaniec, E

    H. Iwaniec, E. Kowalski,Analytic Number Theory.Amer. Math. Soc., (2004)

    2004

  10. [18]

    Koymans, R

    P. Koymans, R. Paterson, T. Santens, A. Shute,Local solubility of generalised Fermat equations, arXiv, (2025), https://arxiv.org/abs/2501.17619

    2025 arXiv

  11. [19]

    Loughran, The number of varieties in a family which contain a rational point, J

    D. Loughran, The number of varieties in a family which contain a rational point, J. Euro. Math. Soc., 20(10), pp.2539–2588, (2018)

    2018

  12. [20]

    Loughran, N

    D. Loughran, N. Rome, E. Sofos,The leading constant for rational points in families, arXiv, (2022),https: //arxiv.org/abs/2210.13559

    2022 arXiv

  13. [21]

    Loughran, A

    D. Loughran, A. Smeets,Fibrations with few rational points, Geom. Func. Anal.26(5), pp.1449–1482, (2016)

    2016

  14. [22]

    Loughran, R

    D. Loughran, R. Takloo-Bighash, S. Tanimoto,Zero-loci of Brauer group elements on semi-simple algebraic groups, Journal de l’Institut de Mathématiques de Jussieu,19(5), pp.1467-1507, (2020)

    2020

  15. [23]

    Peyre,Liberté et accumulation, Documenta Math.25, pp

    E. Peyre,Liberté et accumulation, Documenta Math.25, pp. 1615–1659, (2017)

    2017

  16. [24]

    E. S. Selmer,The Diophantine equationax3 + by3 + cz3 = 0, Acta. Math.85, pp.203-362, (1951)

    1951

  17. [25]

    , Tn)), C

    , Spécialisation des éléments de Br2(Q(T1, . . . , Tn)), C. R. Acad. Sci. Paris Sér. I Math. 311(7), pp.397-402, (1990)

    1990

  18. [26]

    Shiu,A Brun-Titchmarsh theorem for multiplicative functions,J

    P. Shiu,A Brun-Titchmarsh theorem for multiplicative functions,J. reine angew. Math.313, pp.161-170, (1980)

    1980

  19. [27]

    Sofos,Serre’s problem on the density of isotropic fibres in conic bundles, Proc

    E. Sofos,Serre’s problem on the density of isotropic fibres in conic bundles, Proc. of the Lond. Math. Soc. 113(2), (2016), https://doi.org/10.1112/plms/pdw030

    2016 doi

  20. [28]

    Sofos, E

    E. Sofos, E. Visse-Martindale,The density of fibres with a rational point for a fibration over hyper- surfaces of low degree, Annales de l’Institut Fourier,71(2), pp.679-709, (2021)

    2021

  21. [29]

    Soundararajan, Nonvanishing of Quadratic Dirichlet L-Functions at s = 1/2, Annals of Mathematics, 152(2), pp.447-488, (2000)

    K. Soundararajan, Nonvanishing of Quadratic Dirichlet L-Functions at s = 1/2, Annals of Mathematics, 152(2), pp.447-488, (2000)

    2000

  22. [30]

    Wilson, General bilinear forms in the Jacobi Symbol over hyperbolic regions, Monatsh

    C. Wilson, General bilinear forms in the Jacobi Symbol over hyperbolic regions, Monatsh. Math., 202, pp.435-451, (2023), https://doi.org/10.1007/s00605-023-01848-9

    2023 doi

  23. [31]

    C. Wilson,Asymptotics for local solubility of diagonal quadrics over a split quadric surface, ArXiV, (2024), https://arxiv.org/abs/2404.11489 Department of Mathematics, University of Glasgow, G12 8QQ United Kingdom Email address: c.wilson.6@research.gla.ac.uk 39

    2024 arXiv

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.