Pith. sign in

REVIEW 3 major objections 8 minor 1 cited by

On the second moment and non-vanishing of central values of Hecke $L$-functions of $r$-th order characters

T0 review · 3 major / 8 minor · reviewed 2026-07-30 · grok-4.5

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

Pith's one-line read A positive share of central values of r-th order Hecke L-functions do not vanish, for every r ≥ 3.

desk verdict Uniform MDS proof that finally gives second-moment asymptotics and positive-proportion non-vanishing for all r≥3, including the first results for r≥5 and for power-free families. read the letter →

arxiv 2607.27131 v1 pith:25Q6TTJH submitted 2026-07-29 math.NT

Adrian Diaconu , Bogdan Ion , Vicenţiu Paşol , Alexandru A. Popa This is my paper · ORCID
classification math.NT MSC 11M4111F6611R42
keywords HeckeL-functionsr-thordercharacterssecondmomentnon-vanishingmultipleDirichletseriesmollificationlargesieveKubota
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 proves asymptotic formulas for the first and second twisted moments of Hecke L-functions attached to r-th order residue symbols, over number fields that contain the 2r-th roots of unity. From those formulas it concludes that a positive proportion of the central values L(1/2, χ_a) are nonzero, both when a runs over square-free ideals and when a runs over r-th power-free ideals. The argument works uniformly for every r ≥ 3 and recovers earlier cubic and quartic results while giving the first such asymptotics for r ≥ 5. The method builds multiple Dirichlet series from Gauss sums, continues them meromorphically by functional equations and large-sieve bounds, and then mollifies the moments. A sympathetic reader cares because non-vanishing of central values is a classical, stubborn problem for higher-order characters, and the paper supplies unconditional density statements that previously existed only for r = 2, 3, 4.

What carries the argument

Weyl-group multiple Dirichlet series built from Kubota series of r-th order Gauss sums. After a sieving step that isolates square-free or r-th power-free ideals, functional equations and convexity bounds produce meromorphic continuation far enough left of the critical line that a standard mollifier extracts a positive non-vanishing density.

What would settle it

Improve or disprove the large-sieve bound for the second-moment sum over pairs of r-th order characters; any improvement past the current exponent immediately enlarges the region of continuation and either captures the conjectural secondary main term or raises the proven non-vanishing proportion.

Watch

Extended reading notes

Core claim

For every integer r ≥ 3 and every number field F containing the 2r-th roots of unity, a positive proportion of the central values L(1/2, χ_a) are nonzero as a runs through square-free ideals of bounded norm (at least roughly 1/12 − ε when r = 3 and 1/(4r + 2) − ε when r ≥ 4) and likewise through r-th power-free ideals ordered by norm. These densities follow from explicit asymptotic formulas, with power-saving error terms, for the twisted first and second moments of the same L-functions.

Load-bearing premise

The power-saving error that lets the mollifier work rests on the existing large-sieve inequality for r-th order residue symbols; if that sieve is weaker, the admissible mollifier length shrinks and the positive-proportion claim can fail.

Editorial extensions

If this is right

  • Unconditional positive-density non-vanishing now holds for every order r ≥ 3, not merely for quadratic, cubic and quartic characters.
  • The same moment asymptotics apply verbatim to the r-th power-free family and give sharper error terms and secondary main terms for small r.
  • The method supplies twisted moments ready for further applications such as one-level density or low-lying zero statistics.
  • Function-field analogues of the same statements become available at once and can exploit the Riemann hypothesis to capture secondary terms already for cubic characters.

Reading between the lines

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

  • The bottleneck identified in the cubic large sieve suggests that any future improvement of Heath-Brown-type inequalities for higher-order characters would automatically upgrade both the error terms and the non-vanishing proportions obtained here.
  • Because the construction is uniform in the global field, the same densities should hold over rational function fields once the corresponding Kubota series are inserted, giving a clean comparison between number-field and function-field non-vanishing.
  • The secondary main term visible for small r is expressed in terms of Whittaker–Fourier coefficients of metaplectic theta functions; progress on those coefficients for r ≥ 4 would make the secondary term fully explicit.
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

3 major / 8 minor

Summary. The paper establishes asymptotic formulas, with power-saving error terms, for the twisted first and second moments of Hecke L-functions attached to r-th order residue symbols (r≥3) over any number field F containing μ_{2r}, for both the square-free and the r-th power-free families (Theorems A, B, D, E and their twisted forms 10.5.2/10.11.2). Via Soundararajan-style mollification it deduces positive-proportion nonvanishing at s=1/2: at least 1/12−ε (r=3) and 1/(4r+2)−ε (r≥4) in the square-free family (Theorem C), and explicit proportions in the r-th power-free family ordered by norm (Theorem F). The method is the multiple Dirichlet series machinery of [Dia04, Dia19, DW21]: a "perfect" MDS is built from Kubota series (§4–5), the square-free/r-th-power-free sieve is implemented by Möbius inversion over series Z(s;h) (§6), convexity in the h-aspect is obtained by composing functional equations (§7), the continuation region follows from the Blomer–Goldmakher–Louvel large sieve (§8, Appendix), residues are computed explicitly (§9), and mollification is carried out in §10. For r=3,4 the results recover (and for r=4 improve) the recent cubic/quartic nonvanishing theorems; for r≥5 the second-moment asymptotic and the nonvanishing proportion are new.

Significance. If correct, this is a substantial advance: the first second-moment asymptotics with power saving for r-th order character families for all r≥5, the first unconditional positive-proportion nonvanishing for all r≥3, and the first results of either kind for the r-th power-free family even at r=3,4. The method is uniform in r and in the field, a genuine conceptual advantage over the ad hoc cubic/quartic arguments. The paper ships several verifiable strengths: fully explicit Euler products for all leading constants (Eqs. (26)–(27), (42)), consistency checks against [CFK+05] at r=2 and against [DdFDS24, CdFD26] at r=3,4 (Remark 9.2.4), and explicit, checkable numerics for the nonvanishing proportions (§10.13, including the bound α_r ≥ 1−ζ(3)/r²). The honest identification of the (MN)^{2/3} large-sieve barrier as the obstruction to secondary terms (Remark 1.1.3), in line with [DFDH26], adds credibility.

major comments (3)
  1. [§7.1.3–7.1.7, proof of Prop. 7.2.2] The h-aspect exponent (r−1)(1−σ) in Prop. 7.2.2 is the single internally derived number on which δ_κ (§8.1.3–8.1.4), the twisted rad(b)-exponent (10.2.1), the mollifier length θ_κ (10.6.1), and hence every quantitative claim in Theorems A–F depend linearly. Its derivation suppresses factors under the notation "≈" across a three-stage functional-equation chain, and the suppressed objects include the coefficients C(φ,η,t;s_3), C'(φ,η,t;s_3) of Props. 5.7.2/5.7.3, whose stated bound ≪|h|^ε at ℜ(s_3)=−ε is not proved independently but deferred to "the results of this section." Please (i) state precisely which factors are dropped at each step of §7.1.3–7.1.7 and prove they are uniformly O(|h|^ε), and (ii) add a sentence confirming that in the worst case Ic with k_v=1 (Table 1) it is the k=1 case of Lemma 4.2.2 — which, unlike k≥2, has no secondary C-term — that limits the local contribution t
  2. [§10.6.1, Eq. (38); §10.12.2] The admissible mollifier length θ_κ = (1−δ̃_κ)/(1+(2r−1)(1−δ̃_κ)) is derived under the hypothesis λ_b ≪ |b|^{−1+ε}, and the value of θ_κ enters the proportions in Theorems C and F directly (§10.13.1: proportion = (C_κ²E_κ/D_κ)ζ_F^S(n_κ)·θ_κ/(θ_κ+1)−ε'). The eventual choice of λ_κ(b) in §10.12.2 is only asserted to "satisfy the same growth estimates" as in [DdFDS24, Sec. 9]. Since the verification for general r involves the multiplicative functions G_κ, H_κ built from the polynomials P^{(κ)}_r, please include the short computation confirming λ_κ(b) ≪ |b|^{−1+ε} (and H_κ(p_v)>0, which is used for the positivity of E_κ).
  3. [§8.2.2, Prop. 8.2.2 (r=3 improvement)] The improvement A_3=1/3 rests on the bound S_ψ(σ)≪_ε Σ |e|^{2−4σ}|L(1/2,ψχ_{ae²})|²/|a|^{1+ε} and its dyadic analysis using [DdFDS24, Prop. 4.2] "or its version for a general field F in Theorem A.3.4." However, Appendix A.3.4 as stated bounds Σ_{q∈F(Q_1,Q_2)} |L(1/2,ψχ_{qh})|² with the family split as q=q_sf q_full, which is not literally the sum Σ_{q_1≍Q_1, q_2≍Q_2}|L(1/2,ψχ_{q_1 q_2² e²})|² with the e^{1/3}-dependent threshold used in the displayed dyadic estimate. Please spell out exactly how Cor. A.3.4 specializes to the inner sum in the proof of Prop. 8.2.2, including the comparison between the |e|^{1/3} and |e|^{1/2} terms under the restriction Q_1/Q_2>|e|^{1/3}.
minor comments (8)
  1. [Abstract / §1.1] The abstract claims results "over global fields," but the body proves the number-field case only, with function-field analogues merely sketched (§1.3). Either soften the abstract or state precisely which statements are proved in the function-field setting.
  2. [§1.3] "In light of the discussion in Theorem 1.1.3" should refer to Remark 1.1.3.
  3. [§2.1 and §10.1.1] The notation â := (rad a)^r/a (for (r+1)-th power-free a) is easy to confuse with the running ideal variable a and with a_0, a_1; similarly b in §10.1.1. Consider a distinct letter and a displayed definition.
  4. [§7.2.2, proof, Case II] The paragraph after Table 2 contains a broken sentence ("...is offset by the negative power of q_v coming from §7.1.6. we have ord_v f_e = l_v so ord_v j = 0..."). Please repair the flow and clarify which subcase of IIb is being discussed.
  5. [§9.0.2] The principal-part formula (A(1/2), (A'(1/2)+B'(1/2))/2) is justified only by "expressing the limit as a double limit in two ways." A two-line computation or a reference (e.g., to [DGH03] or [DW21]) would help.
  6. [§3.2.3–3.2.5] In the definition of γ(ψ_E) (Eq. (6)), note explicitly that the value is independent of the chosen dataset Δ_E (it follows from Lemma 3.2.5 and triviality of ψ_E on S-units), since γ is used in the definition of τ (Eq. (7)) throughout §4–5.
  7. [References] Several key citations are very recent preprints ([DdFDS24], [CdFD26], [DFDH26], [Ham26], [DMP+]). Please update with stable publication data where available, and double-check that [DdFDS24, Prop. 4.2] and [BGL14, Thm. 1.3] are cited with the exact statements used.
  8. [§1.1.5, Theorem C] It would be helpful to state the r=3 proportion of [DdFDS24] numerically next to 1/12−ε so the reader can gauge the cost of not optimizing over Y; currently only a qualitative remark is given.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: standard MDS-to-moments-to-mollified-nonvanishing chain with independent external and prior lemmas.

full rationale

The derivation is the classical analytic-number-theory pipeline: construct Weyl-group multiple Dirichlet series from Kubota series and Gauss sums (building on Dia04 and BB06), obtain functional equations and convexity bounds, sieve to the square-free / r-th-power-free families, continue past the polar line using the external large sieve of Blomer–Goldmakher–Louvel, extract residues for twisted first and second moments, then mollify exactly as in Soundararajan / DdFDS24 / CdFD26. Self-citations (Dia04, Dia19, DW21) supply prior MDS technology whose hypotheses do not include the target non-vanishing proportions for general r≥3; they function as independent lemmas. There is no fitted parameter renamed as a prediction, no uniqueness theorem imported to forbid alternatives, no ansatz smuggled in that already assumes the conclusion, and no self-definitional loop (moments are not defined in terms of non-vanishing, nor vice versa). Internal bookkeeping of the h-aspect convexity exponent is a correctness/fragility question, not circularity. Score 0 is the honest finding.

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

The central nonvanishing claim rests on standard analytic number theory (functional equations, Phragmén–Lindelöf, Mellin inversion, Tauberian extraction of main terms), on two external analytic inputs (large sieve for r-th order characters; meromorphic continuation of Kubota series), and on the standing arithmetic hypothesis that F contains μ_{2r} so that [BB06] applies directly. Mollifier length and test function W are free choices within ranges forced by the error exponents; they are not fitted to data. No new physical or arithmetic entities are postulated.

free parameters (3)
  • mollifier length exponent θ_κ = θ_1 = (1−δ_1)/(1+(2r−1)(1−δ_1)); for r=3, θ_1=1/11
    Chosen as large as the error term in the twisted second-moment asymptotic permits (eq. (38)); controls the nonvanishing proportion θ_κ/(θ_κ+1). Not fitted to data—forced by convexity/large-sieve abscissa.
  • Schwartz test function W
    Smooth compactly supported weight on (1,2) used to smooth the moment sums; leading coefficients involve its Mellin transform c_W. Arbitrary within the stated class; final proportions take W→1_{(1,2)}.
  • mollifier coefficients λ_κ(b) / ξ_κ(b)
    Explicitly constructed from Euler factors G_κ, H_κ in §10.12 following the Soundararajan/DdFDS24 template; not numerically fitted.
assumptions (5)
  • domain assumption F contains the group μ_{2r} of 2r-th roots of unity (so that Brubaker–Bump [BB06, Thm. 1] applies directly to Kubota series).
    Stated in §1.1.1 and Remark 1.4.1; authors note the restriction is for convenience and can in principle be removed with more work.
  • domain assumption Large sieve inequality for r-th order residue symbols [BGL14, Thm. 1.3], used to bound sums of |L(1/2,ψχ_af)|^2 (Prop. 7.2.4 and Appendix).
    Load-bearing for the abscissa of continuation of the sieved MDS and thus for power-saving errors; for r=3 the sharper [DdFDS24, Prop. 4.2] is used.
  • domain assumption Meromorphic continuation, functional equation, and convexity of the completed Kubota series eD_S(s,a,ψ) from [BB06].
    §4.1.4; poles only at 1/2±1/r; used to build the perfect MDS and its convexity bounds.
  • standard math Standard functional equations and convexity bounds for Hecke L-functions of trivial infinity type; Phragmén–Lindelöf; Mellin inversion; Landau’s lemma for Dirichlet series with nonnegative coefficients.
    Used throughout §§5–10 for contour shifts and residue extraction.
  • domain assumption O_S is a PID (S chosen large enough) and the Fisher–Friedberg extension of the r-th power residue symbol is a Hecke character of the stated conductor.
    §1.1.1–§3.1; standard setup from [FHL03, FF04].

how reviews work

0 comments
Cite this review

Pith. "Pith review of On the second moment and non-vanishing of central values of Hecke $L$-functions of $r$-th order characters." pith.science (2026). https://pith.science/paper/25Q6TTJH

@misc{pith2026260727131,
  author       = {Pith},
  title        = {Pith review of: On the second moment and non-vanishing of central values of Hecke $L$-functions of $r$-th order characters},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/25Q6TTJH}},
  note         = {Machine review of arXiv:2607.27131}
}
abstract

In this paper, we establish asymptotic formulas for the first and second twisted moments of $r$-th order Hecke $L$-functions over global fields that contain the $2r$-th roots of unity, for $r\ge 3$. We focus primarily on algebraic number fields. As a consequence, we establish a positive proportion of non-vanishing central values for these $L$-functions, specifically for families of both square-free and $r$-th power-free ideals. Our approach is based on the machinery of multiple Dirichlet series.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$

    math.NT 2026-08 conditional novelty 7.0 of 10

    Under GRH the average analytic rank of y²=x³-dx over odd fourth-power-free d is at most 13/6, and at most 3/2 assuming a quartic Gauss-sum conjecture.

Reference graph

Works this paper leans on

106 extracted references · 27 canonical work pages · cited by 1 Pith paper

  1. [1]

    International Mathematics Research Notices , year =

    Blomer, Valentin and Goldmakher, Leo and Louvel, Beno. International Mathematics Research Notices , year =

  2. [2]

    Estimation de sommes multiples de fonctions arithm

    de la Bret. Estimation de sommes multiples de fonctions arithm. Compositio Mathematica , volume =. 2001 , pages =

    2001

  3. [3]

    Journal f

    Brubaker, Ben and Bump, Daniel , title =. Journal f. 2006 , pages =

    2006

  4. [4]

    Multiple Dirichlet Series, Automorphic Forms, and Analytic Number Theory , series =

    Brubaker, Ben and Bump, Daniel and Chinta, Gautam and Friedberg, Solomon and Hoffstein, Jeffrey , title =. Multiple Dirichlet Series, Automorphic Forms, and Analytic Number Theory , series =. 2006 , pages =

    2006

  5. [5]

    Annals of Mathematics , series =

    Brubaker, Ben and Bump, Daniel and Friedberg, Solomon and Hoffstein, Jeffrey , title =. Annals of Mathematics , series =. 2007 , pages =

    2007

  6. [6]

    Algebraic Number Theory , publisher =

  7. [7]

    , title =

    Chinta, Gautam and Gunnells, Paul E. , title =. Journal of the American Mathematical Society , volume =. 2010 , pages =

    2010

  8. [8]

    Brian and Farmer, David W

    Conrey, J. Brian and Farmer, David W. and Keating, Jonathan P. and Rubinstein, Michael O. and Snaith, Nina C. , title =. Proceedings of the London Mathematical Society , series =. 2005 , pages =

    2005

Show all 106 references
  1. [9]

    2024 , eprint =

    David, Chantal and de Faveri, Alessandro and Dunn, Alexander and Stucky, Joshua , title =. 2024 , eprint =

    2024

  2. [10]

    2026 , eprint =

    Castillo, Cruz and de Faveri, Alexandre and Dunn, Alexander , title =. 2026 , eprint =

    2026

  3. [11]

    Inventiones Mathematicae , volume =

    Diaconu, Adrian , title =. Inventiones Mathematicae , volume =. 2004 , pages =

    2004

  4. [12]

    Journal of Number Theory , volume =

    Diaconu, Adrian , title =. Journal of Number Theory , volume =. 2019 , pages =

    2019

  5. [13]

    Journal of the European Mathematical Society , volume =

    Diaconu, Adrian and Whitehead, Ian , title =. Journal of the European Mathematical Society , volume =. 2021 , pages =

    2021

  6. [14]

    Mathematische Annalen , volume =

    Friedberg, Solomon and Hoffstein, Jeffrey and Lieman, Daniel , title =. Mathematische Annalen , volume =. 2003 , pages =

    2003

  7. [15]

    A quadratic large sieve inequality over number fields , journal =

    Goldmakher, Leo and Louvel, Beno. A quadratic large sieve inequality over number fields , journal =. 2013 , pages =

    2013

  8. [16]

    Iwaniec, Henryk , title =

  9. [17]

    Iwaniec, Henryk and Kowalski, Emmanuel , title =

  10. [18]

    Kubota, Tomio , title =

  11. [19]

    and Patterson, S

    Kazhdan, David A. and Patterson, S. J. , title =. Publications Math. 1984 , pages =

    1984

  12. [20]

    Algebraic Number Theory , series =

    Neukirch, J. Algebraic Number Theory , series =

  13. [21]

    Rosen, Michael , title =

  14. [22]

    Annals of Mathematics , volume =

    Soundararajan, Kannan , title =. Annals of Mathematics , volume =. 2000 , pages =

    2000

  15. [23]

    Bias in cubic

    Dunn, Alexander and Radziwi. Bias in cubic. Annals of Mathematics , series =. 2024 , pages =

    2024

  16. [24]

    and Patterson, S

    Eckhardt, C. and Patterson, S. J. , title =. Proceedings of the London Mathematical Society , series =. 1992 , pages =

    1992

  17. [25]

    Brian and Iwaniec, Henryk and Soundararajan, Kannan , title =

    Conrey, J. Brian and Iwaniec, Henryk and Soundararajan, Kannan , title =. Geometric and Functional Analysis , volume =. 2012 , pages =. doi:10.1007/s00039-012-0191-6 , url =

    2012 doi

  18. [26]

    , TITLE =

    Hamdar, Mohammad H. , TITLE =. Math. Ann. , FJOURNAL =. 2026 , NUMBER =. doi:10.1007/s00208-026-03341-8 , URL =

    2026 doi

  19. [27]

    Mathematische Zeitschrift , volume =

    Shen, Quanli , title =. Mathematische Zeitschrift , volume =. 2021 , pages =. doi:10.1007/s00209-020-02609-2 , url =

    2021 doi

  20. [28]

    Nonvanishing of

    Khan, Rizwanur and Mili. Nonvanishing of. Mathematische Zeitschrift , volume =. 2022 , pages =. doi:10.1007/s00209-021-02821-8 , url =

    2022 doi

  21. [29]

    On moments of twisted

    Blomer, Valentin and Fouvry,. On moments of twisted. American Journal of Mathematics , volume =. 2017 , pages =. doi:10.1353/ajm.2017.0019 , url =

    2017

  22. [30]

    David, Chantal and Florea, Alexandra M. and Lal. Nonvanishing of. 2025 , eprint =

    2025

  23. [31]

    Mathematische Annalen , volume =

    Wu, Xiaosheng , title =. Mathematische Annalen , volume =. 2023 , pages =. doi:10.1007/s00208-022-02483-9 , url =

    2023 doi

  24. [32]

    Inventiones Mathematicae , volume =

    Goldfeld, Dorian and Hoffstein, Jeffrey , title =. Inventiones Mathematicae , volume =. 1985 , pages =. doi:10.1007/BF01388603 , url =

    1985 doi

  25. [33]

    Vinogradov, A. I. and Takhtadzhyan, L. A. , title =. Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta imeni V. A. Steklova , volume =. 1981 , pages =

    1981

  26. [34]

    , title =

    Florea, Alexandra M. , title =. Geometric and Functional Analysis , volume =. 2017 , pages =. doi:10.1007/s00039-017-0409-8 , url =

    2017 doi

  27. [35]

    , title =

    Young, Matthew P. , title =. Annals of Mathematics , series =. 2011 , pages =. doi:10.4007/annals.2011.173.1.1 , url =

    2011 doi

  28. [36]

    Inventiones Mathematicae , volume =

    Li, Xiaoqing , title =. Inventiones Mathematicae , volume =. 2024 , pages =

    2024

  29. [37]

    2024 , eprint =

    Shen, Quanli and Stucky, Joshua , title =. 2024 , eprint =

    2024

  30. [38]

    2025 , note =

    Goel, Shivani and Ray, Anwesh , title =. 2025 , note =

    2025

  31. [39]

    Analytic Number Theory , series =

    Soundararajan, Kannan , title =. Analytic Number Theory , series =. 2007 , pages =

    2007

  32. [40]

    , title =

    Florea, Alexandra M. , title =. Forum Mathematicum , volume =. 2017 , pages =. doi:10.1515/forum-2015-0152 , url =

    2017 doi

  33. [41]

    Bui, H. M. and Florea, Alexandra M. , title =. Transactions of the American Mathematical Society , volume =. 2018 , pages =. doi:10.1090/tran/7317 , url =

    2018 doi

  34. [42]

    Hyperelliptic curves, the scanning map, and moments of families of quadratic

    Bergstr. Hyperelliptic curves, the scanning map, and moments of families of quadratic. 2023 , eprint =

    2023

  35. [43]

    , title =

    Florea, Alexandra M. , title =. International Mathematics Research Notices , year =. doi:10.1093/imrn/rnv387 , url =

    doi

  36. [44]

    , title =

    Baier, Stephan and Young, Matthew P. , title =. Journal of Number Theory , volume =. 2010 , pages =. doi:10.1016/j.jnt.2009.11.007 , url =

    2010 doi

  37. [45]

    Indagationes Mathematicae , volume =

    Gao, Peng and Zhao, Liangyi , title =. Indagationes Mathematicae , volume =. 2022 , pages =. doi:10.1016/j.indag.2022.08.003 , url =

    2022 doi

  38. [46]

    and Sarnak, Peter , title =

    Katz, Nicholas M. and Sarnak, Peter , title =. 1999 , doi =

    1999

  39. [47]

    Analysis , volume =

    Jutila, Matti , title =. Analysis , volume =. 1981 , pages =. doi:10.1524/anly.1981.1.2.149 , url =

    1981 doi

  40. [48]

    and Li, Wanlin and Shusterman, Mark , title =

    Ellenberg, Jordan S. and Li, Wanlin and Shusterman, Mark , title =. Algebra & Number Theory , volume =. 2020 , pages =. doi:10.2140/ant.2020.14.1895 , url =

    2020 doi

  41. [49]

    Signs of

    Lester, Stephen and Radziwi. Signs of. Mathematische Annalen , volume =. 2021 , pages =. doi:10.1007/s00208-020-02123-0 , url =

    2021 doi

  42. [50]

    , title =

    Harper, Adam J. , title =. 2013 , note =

    2013

  43. [51]

    Annals of Mathematics , series =

    Soundararajan, Kannan , title =. Annals of Mathematics , series =. 2009 , pages =. doi:10.4007/annals.2009.170.981 , url =

    2009 doi

  44. [52]

    David, Chantal and Florea, Alexandra M. and Lal. The mean values of cubic. Algebra & Number Theory , volume =. 2022 , pages =. doi:10.2140/ant.2022.16.1259 , url =

    2022 doi

  45. [53]

    David, Chantal and Florea, Alexandra M. and Lal. Nonvanishing for cubic. Forum of Mathematics, Sigma , volume =. 2021 , pages =. doi:10.1017/fms.2021.62 , url =

    2021 doi

  46. [54]

    Number Theory in Progress, Vol

    Iwaniec, Henryk and Sarnak, Peter , title =. Number Theory in Progress, Vol. 2 (Zakopane--Ko. 1999 , pages =

    1999

  47. [55]

    and Murty, V

    Balasubramanian, R. and Murty, V. Kumar , title =. Annales Scientifiques de l'. 1992 , pages =

    1992

  48. [56]

    Acta Arithmetica , volume =

    On the distribution of the nontrivial zeros of quadratic. Acta Arithmetica , volume =. 1999 , pages =. doi:10.4064/aa-91-3-209-228 , url =

    1999 doi

  49. [57]

    Chowla, Sarvadaman , title =

  50. [58]

    Compositio Mathematica , volume =

    Luo, Wenzhi , title =. Compositio Mathematica , volume =. 2004 , pages =. doi:10.1112/S0010437X0400051X , url =

    2004 doi

  51. [59]

    Non-vanishing of cubic

    G. Non-vanishing of cubic. Proceedings of the American Mathematical Society , volume =. 2025 , pages =

    2025

  52. [60]

    One-level density and non-vanishing for cubic

    David, Chantal and G. One-level density and non-vanishing for cubic. International Mathematics Research Notices , year =. doi:10.1093/imrn/rnab240 , url =

    doi

  53. [61]

    Mollified moments of cubic

    G. Mollified moments of cubic. Journal of Mathematical Analysis and Applications , volume =. 2024 , pages =. doi:10.1016/j.jmaa.2023.128014 , url =

    2024

  54. [62]

    Heath-Brown, D. R. , title =. Analysis , volume =. 1981 , pages =. doi:10.1524/anly.1981.1.1.25 , url =

    1981 doi

  55. [63]

    Compositio Mathematica , volume =

    Diaconu, Adrian and Goldfeld, Dorian and Hoffstein, Jeffrey , title =. Compositio Mathematica , volume =. 2003 , pages =. doi:10.1023/B:COMP.0000018137.38458.68 , url =

    2003

  56. [64]

    Journal f

    Suzuki, Toshiaki , title =. Journal f. 1983 , pages =

    1983

  57. [65]

    Heath-Brown, D. R. , title =. Acta Arithmetica , volume =. 1995 , pages =. doi:10.4064/aa-72-3-235-275 , url =

    1995 doi

  58. [66]

    Hasse, Helmut , title =

  59. [67]

    Heath-Brown, D. R. , title =. Israel Journal of Mathematics , volume =. 2000 , pages =. doi:10.1007/s11856-000-1273-y , url =

    2000 doi

  60. [68]

    Theta Functions: From the Classical to the Modern , editor =

    Hoffstein, Jeffrey , title =. Theta Functions: From the Classical to the Modern , editor =. 1993 , pages =

    1993

  61. [69]

    Patterson, S. J. , title =. Journal f. 1977 , pages =. doi:10.1515/crll.1977.296.125 , url =

    1977 doi

  62. [70]

    1998 , doi =

    Proskurin, Nikolai , title =. 1998 , doi =

    1998

  63. [71]

    Heath-Brown, D. R. and Patterson, S. J. , title =. Journal f. 1979 , pages =

    1979

  64. [72]

    Patterson, S. J. , title =. Journal f. 1977 , pages =. doi:10.1515/crll.1977.296.217 , url =

    1977 doi

  65. [73]

    Patterson, S. J. , title =. Journal f. 1982 , pages =. doi:10.1515/crll.1982.336.185 , url =

    1982 doi

  66. [74]

    Publications Math

    Deligne, Pierre , title =. Publications Math. 1974 , pages =

    1974

  67. [75]

    Publications Math

    Deligne, Pierre , title =. Publications Math. 1980 , pages =

    1980

  68. [76]

    2023 , eprint =

    David, Chantal and Meisner, Patrick , title =. 2023 , eprint =

    2023

  69. [77]

    and Keating, Jonathan P

    Andrade, Julio C. and Keating, Jonathan P. , title =. Journal of Number Theory , volume =. 2014 , pages =

    2014

  70. [78]

    Annales Scientifiques de l'

    Borel, Armand , title =. Annales Scientifiques de l'. 1974 , pages =

    1974

  71. [79]

    Manifolds and Lie Groups (Notre Dame, Ind., 1980) , series =

    Borel, Armand , title =. Manifolds and Lie Groups (Notre Dame, Ind., 1980) , series =. 1981 , pages =

    1980

  72. [80]

    Bui, H. M. and Florea, Alexandra M. and Keating, Jonathan P. and Roditty-Gershon, Edva , title =. Algebra & Number Theory , volume =. 2020 , pages =

    2020

  73. [81]

    Brian and Farmer, David W

    Conrey, J. Brian and Farmer, David W. , title =. International Mathematics Research Notices , year =

  74. [82]

    2024 , eprint =

    David, Chantal and Devin, Lucile and Waxman, Ezra , title =. 2024 , eprint =

    2024

  75. [83]

    Annals of Mathematics , series =

    Diaconu, Adrian and Tian, Ye , title =. Annals of Mathematics , series =. 2005 , pages =

    2005

  76. [84]

    Journal of Number Theory , volume =

    Diaconu, Adrian and Twiss, Henry , title =. Journal of Number Theory , volume =. 2023 , pages =

    2023

  77. [85]

    Quadratic

    Diaconu, Adrian and Pa. Quadratic. Amer. J. Math. , FJOURNAL =. 2025 , NUMBER =. doi:10.1353/ajm.2025.a971089 , URL =

    2025 doi

  78. [86]

    Duke Mathematical Journal , volume =

    Fisher, Benji and Friedberg, Solomon , title =. Duke Mathematical Journal , volume =. 2003 , pages =. doi:10.1215/S0012-7094-03-11735-4 , url =

    2003 doi

  79. [87]

    Compositio Mathematica , volume =

    Fisher, Benji and Friedberg, Solomon , title =. Compositio Mathematica , volume =. 2004 , pages =

    2004

  80. [88]

    , title =

    Florea, Alexandra M. , title =. International Mathematics Research Notices , year =

  81. [89]

    2023 , eprint =

    Hoang, Anh Tuan Nam , title =. 2023 , eprint =

    2023

  82. [90]

    Journal f

    Hoffstein, Jeffrey and Rosen, Michael , title =. Journal f. 1992 , pages =. doi:10.1515/crll.1992.426.117 , url =

    1992 doi

  83. [91]

    , title =

    Katz, Nicholas M. , title =

  84. [92]

    and Sarnak, Peter , title =

    Katz, Nicholas M. and Sarnak, Peter , title =. Bulletin of the American Mathematical Society , series =. 1999 , pages =

    1999

  85. [93]

    and Snaith, Nina C

    Keating, Jonathan P. and Snaith, Nina C. , title =. Communications in Mathematical Physics , volume =. 2000 , pages =

    2000

  86. [94]

    2024 , eprint =

    Miller, Jeremy and Patzt, Peter and Petersen, Dan and Randal-Williams, Oscar , title =. 2024 , eprint =

    2024

  87. [95]

    Advances in Mathematics , volume =

    Randal-Williams, Oscar and Wahl, Nathalie , title =. Advances in Mathematics , volume =. 2017 , pages =

    2017

  88. [96]

    and Wu, Kaiyu , title =

    Rubinstein, Michael O. and Wu, Kaiyu , title =. Philosophical Transactions of the Royal Society A , volume =. 2015 , pages =

    2015

  89. [97]

    Acta Arithmetica , volume =

    Rudnick, Zeev , title =. Acta Arithmetica , volume =. 2010 , pages =

    2010

  90. [98]

    Algebra & Number Theory , volume =

    Sawin, Will , title =. Algebra & Number Theory , volume =. 2020 , pages =

    2020

  91. [99]

    Duke Mathematical Journal , volume =

    Sawin, Will , title =. Duke Mathematical Journal , volume =. 2021 , pages =

    2021

  92. [100]

    Acta Math

    Sawin, Will , TITLE =. Acta Math. , FJOURNAL =. 2024 , NUMBER =. doi:10.4310/acta.2024.v233.n2.a3 , URL =

    2024 doi

  93. [101]

    Venkataramana, T. N. , title =. Annals of Mathematics , series =. 2014 , pages =

    2014

  94. [102]

    , title =

    Wang, Victor Y. , title =. 2024 , eprint =

    2024

  95. [103]

    , title =

    Young, Matthew P. , title =. Selecta Mathematica , series =. 2013 , pages =

    2013

  96. [104]

    2025 , eprint =

    Hong, Ziwei and Zheng, Zhiyong , title =. 2025 , eprint =. doi:10.48550/arXiv.2506.14656 , url =

    2025 doi arXiv

  97. [105]

    Diaconu, Adrian and Miller, Jeremy and Patzt, Peter and Petersen, Dan and Wang, Victor , title =

  98. [106]

    2026 , eprint =

    De Faveri, Alexandre and Dunn, Alexander and Hoffstein, Jeffrey , title =. 2026 , eprint =

    2026

Pith tools

Reviewed July 30, 2026 · model on record in the stance chip above.