{"id":"05246acb-95b5-47a3-9c0c-3b1a7436521a","arxiv_id":"2607.27131","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":3,"one_line_summary":"Asymptotics for twisted first and second moments of r-th order Hecke L-functions yield a positive proportion of non-vanishing central values for square-free and r-th power-free ideal families when r≥3.","lead":"The paper proves asymptotic formulas for the first and second moments of r-th order Hecke L-functions (r≥3) over number fields containing the 2r-th roots of unity, and deduces that a positive proportion of these L-functions do not vanish at the central point. It gives the first such second-moment asymptotics for r≥5 and new results even for power-free families.","discovery_kind":"extension","skeptic_critique":{"model":"moonshotai/kimi-k3","headline":"Every quantitative theorem (A–F) inherits its error term, mollifier length, and nonvanishing proportion from one internally derived number: the h-aspect exponent (r−1)(1−σ) in the convexity bound Thm. 7.2.2, whose proof suppresses factors via \"≈\" across a three-stage functional-equation chain.","rationale":"I partially agree with the reader: the BGL large sieve and BB06 convexity are indeed the analytic inputs on which the continuation region rests, but they are external, published, and — per [DFDH26] — their exponents are known to be best possible in the cubic case, so they are load-bearing but not fragile. The more genuinely soft spot is one step downstream: the internal convexity bookkeeping of §7 that turns those inputs into the h-exponent (r−1)(1−σ), because (i) it is new with this paper, (ii) its proof explicitly suppresses factors (\"≈\") through a three-stage functional-equation chain, and (iii) every headline quantity is a monotone function of that exponent. I nonetheless leave the verdict at ACCEPT for three reasons. First, the machinery is cross-validated in the two cases where independent proofs exist: the paper recovers [DdFDS24]'s 5/6 barrier for r=3 (§8.2) and improves [CdFD26] for r=4, and the exponent chain has the same form for all r. Second, I spot-checked the downstream arithmetic and it is internally consistent: 1−δ_1=1/(2(r+1)) yields θ_1=1/(4r+1) and hence the proportion 1/(4r+2) of Theorem C; δ_0=(3r−2)/(4r−2) yields the proportion r/(2r²+4r−2) in Theorem F; the regions capture the secondary pole at w=1/2+1/r exactly in the advertised ranges (r=3 for κ=1; r≤6 first moment, r=3,4 second moment for κ=0). Third, the conclusion degrades gracefully: a small error in the exponent would weaken error terms and proportions before it destroyed positivity, so the qualitative headline (positive proportion for all r≥3) is more robust than the quantitative one. The proposed test is cheap relative to the paper's length and would settle the concern decisively.","tokens_in":67580,"tokens_out":7793,"duration_ms":864196,"concrete_test":"Re-derive the worst-case q_v-exponent in Thm. 7.2.2 with all \"≈\"-suppressed factors made explicit, for one prime v in Case Ic with k_v=1: start from Ẑ^{S∪S_h}(1/2,1/2,s_3;χ_f,χ_f,ρχ_g) with ℜ(s_3)=−ε, apply in sequence Prop. 5.7.2, Cor. 4.2.4, Thm. 4.1.4, Cor. 4.2.6, and Prop. 5.7.3, recording every Euler factor, Gauss-sum normalization, |g_1|^{1/2−s_3} and |j_1|^{1/2−s''_3} conductor term, and confirm the total is q_v^{r−1+ε} and not q_v^{r+ε}. Corroborate numerically for r=3, F=Q(√−3): compute Σ_{a sq-free,|a|≤X}|L(1/2,χ_a)|² for growing X and check the remainder after XR^{(1)}_W(log X) is consistent with X^{5/6+ε} (matching the independent [DdFDS24] proof), not X^{11/12}.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader located the weakest assumption in the external inputs (BGL large sieve, BB06 Kubota theory). Those are published, standard, and — as the paper itself notes via [DFDH26] — the (MN)^{2/3} barrier in the large sieve is provably intrinsic, so no hidden fragility there. The point I find least secure is internal: the conversion of those inputs into the continuation region. Specifically, Theorem 7.2.2 asserts the h-exponent (r−1)(1−σ)+ε for Ẑ^{S∪S_h}(1/2,1/2,s_3;·), derived in §7.1.3–7.1.7 by composing σ_3, the Kubota functional equation (Thm. 4.1.4), the D^S↔D^T transfers (Cors. 4.2.4/4.2.6), and Prop. 5.7.3, while explicitly dropping factors under the notation \"≈\" and collecting q_v-powers in Tables 1–2. The worst case (Ic, k_v=1) gives q_v^{r−1+ε}. Everything downstream is a linear function of this exponent: δ_1=(r+A_r)/(r+1) (§8.1.3), δ_0=(3r−2)/(4r−2) (§8.1.4), the twisted-uniform rad(b)-exponent (10.2.1), the mollifier length θ_κ (10.6.1), and hence the proportions 1/(4r+2) and the α_r-formulas in Theorems C and F. If the true worst-case local contribution were q_v^r rather than q_v^{r−1} (e.g., a conductor factor |f_1|^{1/2−s_3} or a C(φ,η,t;s_3)≪|h|^ε bound mis-estimated at k_v=1), δ_κ would shift upward; for large r, where 1−δ_1=1/(2(r+1)) is already small, even a unit shift in the exponent eliminates the power saving and with it Theorems A–F as stated. The proof's own bookkeeping is the only place this is verified, and it is precisely the kind of estimate where a suppressed factor can hide an extra q_v.","agreement_with_reader":"partial"},"referee_report":{"model":"moonshotai/kimi-k3","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.","tokens_in":67995,"tokens_out":6034,"duration_ms":177078,"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":[{"comment":"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","section":"§7.1.3–7.1.7, proof of Prop. 7.2.2"},{"comment":"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).","section":"§10.6.1, Eq. (38); §10.12.2"},{"comment":"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}.","section":"§8.2.2, Prop. 8.2.2 (r=3 improvement)"}],"minor_comments":[{"comment":"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.","section":"Abstract / §1.1"},{"comment":"\"In light of the discussion in Theorem 1.1.3\" should refer to Remark 1.1.3.","section":"§1.3"},{"comment":"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.","section":"§2.1 and §10.1.1"},{"comment":"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.","section":"§7.2.2, proof, Case II"},{"comment":"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.","section":"§9.0.2"},{"comment":"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.","section":"§3.2.3–3.2.5"},{"comment":"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.","section":"References"},{"comment":"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.","section":"§1.1.5, Theorem C"}],"recommendation":"minor_revision","confidential_remarks":"The paper is long and technically dense, and full verification of §5–7 would require an unusually large time investment; my recommendation rests on the internal consistency checks (exact recovery of the r=3,4 results and the r=2 sanity check), the standard nature of the external inputs, and the plausibility of the case analysis in Tables 1–2. The citation pattern is heavily weighted toward the first author's prior MDS work, but this reflects the actual dependence of the machinery rather than padding; overlap with [DdFDS24, CdFD26] at r=3,4 is openly acknowledged and the r=4 error term is improved. Suitable for the journal if the clarifications in the major comments are supplied."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This is the paper that closes the gap for higher-order Hecke L-functions. Until now we had unconditional positive-proportion non-vanishing only for r=2,3,4 by ad-hoc methods; here Diaconu–Ion–Pașol–Popa give a uniform multiple-Dirichlet-series treatment that works for every r≥3 over any number field containing the 2r-th roots of unity, and they treat both the square-free and the r-th-power-free families at once. The second-moment asymptotic for square-free ideals when r≥5 is new, as are essentially all the power-free results.\n\nWhat they do well is the architecture. They build the perfect MDS from Kubota series, sieve, run the functional equations carefully enough to extract an explicit convexity bound in the h-aspect, compute residues with clean Euler products, then mollify. The error exponents and the leading coefficients are written down explicitly; the appendix supplies the large-sieve input in the form they need. That is real work, not a sketch.\n\nThe soft spots are proportionate. The non-vanishing proportions are weaker than the optimized cubic numbers in DdFDS24 because they deliberately skip the truncated-Möbius mollifier; they say so. Everything quantitative (error terms, mollifier length θ_κ, the 1/(4r+2) proportion) traces back to the local exponent (r-1)(1-σ) in the convexity bound of Theorem 7.2.2. The bookkeeping in §7.1 that produces that exponent uses the usual “≈” suppressions and two tables of q_v-powers; a referee will want to check that the worst-case contribution really is r-1 and not r. I do not see an obvious missing conductor factor, but that page is the only place the claim is verified. The external inputs (BGL large sieve, Brubaker–Bump) are standard and already known to be essentially sharp.\n\nThis is for people who work on moments of L-functions or metaplectic forms. It deserves a serious referee. I would send it out.","headline":"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.","tokens_in":69301,"tokens_out":599,"would_cite":true,"duration_ms":18413,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M41","11F66","11R42"],"pacs":[],"model":"grok-4.5","headline":"A positive share of central values of r-th order Hecke L-functions do not vanish, for every r ≥ 3.","keywords":["Hecke L-functions","r-th order characters","second moment","non-vanishing","multiple Dirichlet series","mollification","large sieve","Kubota series"],"falsifier":"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.","tokens_in":68752,"feed_emoji":"∫","tokens_out":1020,"duration_ms":19527,"temperature":0.7,"pith_summary":"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.","feed_headline":"Positive share of r-th order L-values stay nonzero","feed_subtitle":"Moment asymptotics and mollifiers give unconditional density for every order r ≥ 3","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["Positive density of nonvanishing r-th order Hecke L-values","Moments give nonzero central values for r≥3 Hecke L-functions","Positive share of square-free Hecke L-values nonzero at s=1/2","Power-saving moments imply nonvanishing for every r≥3","1/(4r+2) density of nonvanishing r-th order central L-values"],"cache_read_input_tokens":65664,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Positive density of nonvanishing r-th order Hecke L-values","Moments give nonzero central values for r≥3 Hecke L-functions","Positive share of square-free Hecke L-values nonzero at s=1/2","Power-saving moments imply nonvanishing for every r≥3","1/(4r+2) density of nonvanishing r-th order central L-values"]},"model":"grok-4.5","effort":"low","cost_usd":0.003954,"raw_usage":{"total_tokens":1160,"prompt_tokens":697,"num_sources_used":0,"completion_tokens":90,"cost_in_usd_ticks":39544000,"prompt_tokens_details":{"text_tokens":697,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":373,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":697,"tokens_out":90,"duration_ms":6516,"temperature":1.0,"reasoning_tokens":373,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-30T11:59:31.806474+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"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.","supporting_citations":[],"review_version":1}