{"id":"923e3c98-9e5b-4ab2-b812-6cb4d5488f1c","arxiv_id":"2507.06609","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Under GRH, in every Galois orbit modulo p^k, a positive proportion of characters chi satisfy L(1/2, chi eta1) L(1/2, chi eta2) != 0 for distinct imprimitive twists eta1, eta2.","lead":"This paper proves that, assuming the Generalized Riemann Hypothesis, a positive proportion of Dirichlet characters in any fixed Galois orbit have two twisted central L-values simultaneously nonzero, for prime power moduli. It also computes unconditional second moment asymptotics for full and thin Galois orbits.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The reduction of the off-diagonal sums in Theorem 1.5 to the D0 congruence a^{p-1}≡b^{p-1} does not follow from Lemmas 2.3/2.5; the correct support condition is (ab)^{p-1}≡1.","rationale":"The reader identified GRH as the weakest assumption and set a CONDITIONAL verdict pending a fuller proof of Proposition 4.2. My concern is different and more foundational: the unconditional twisted-moment theorem 1.5, used to prove Theorem 1.2 and then Theorem 1.1, appears to contain an invalid reduction of the orbit-average support to the D0 congruence. If the support condition from Lemma 2.3 is (m1m2n1n2)^{p-1}≡1, then the D0 sums over a^{p-1}≡b^{p-1} do not control the terms that actually occur, and the error estimates in Theorem 1.5, Corollary 1.6, and Theorem 1.2 are not justified. This is not a disagreement with the GRH framework or with the use of Roth-Ridout type inputs; it is an internal consistency check in the core computation. I therefore recommend that the verdict move from CONDITIONAL to UNVERDICTED pending the concrete support-set check. If the check confirms the paper's reduction, the CONDITIONAL verdict can be restored; if it confirms the mismatch, the main theorem is unproved as written.","tokens_in":32671,"tokens_out":45342,"duration_ms":538218,"concrete_test":"Re-derive the reduction in §3.1 from Lemma 2.3 with the exact support condition: enumerate (m1,n1,m2,n2) with (m1m2n1n2)^{p-1}≡1 mod p^{k-1}, and compare this set with the D0 condition (m1n1)^{p-1}≡(m2n2)^{p-1} for a sample, e.g. p=3, k=4, dyadic A,B. Then recompute E+(ϑ) using the correct product-congruence sum and check whether the bound O(q^{-1/4+ε}) in (17) still holds for all allowed h. If the supports differ, Theorem 1.5 is not established by the written proof; if the supports actually coincide inside the relevant ranges, the concern is resolved.","verdict_should_be":"UNVERDICTED","load_bearing_attack":"In §3.1, the bound for E+(ϑ) is stated after Lemma 2.3 as a sum over (m1n1)^{p-1}≡(m2n2)^{p-1} mod p^{k-1}, and the bound for E−(ϑ) is similarly derived from Lemma 2.5 using (m1n2)^{p-1}≡(m2n1)^{p-1}. But Lemma 2.3 gives EO[χ(n)]≠0 only when n^{p-1}≡1 mod p^{k-1}. With n=m1m2n1n2 and a=m1n1, b=m2n2, the valid support is (ab)^{p-1}≡1, i.e. a^{p-1}≡b^{-(p-1)}, not a^{p-1}≡b^{p-1}. These two conditions are not equivalent: for p=3, α=3, a=2, b=14 one has ab≡1 mod 27, so (ab)^2≡1, but a^2=4 and b^2≡7 mod 27. Thus D0(A,B;p^α), as defined in (6), neither contains nor clearly bounds the actual nonzero orbit-average contributions. The same issue affects E−, where the product condition is (m1m2n1n2)^{p-1}≡1 mod p^{k-h}. Since Theorem 1.5 is the unconditional basis for Theorem 1.2 and hence for Theorem 1.1, this is a load-bearing gap. The apparent 'diagonal' main term (14) also appears to replace the orbit-average support condition by an unrestricted L(1+2s,η) integral; if that is intentional, the argument needs to show why the missing support condition does not change the residue.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims that, under GRH, for a fixed odd prime p and q=p^k, given any two distinct imprimitive Dirichlet characters eta1, eta2 modulo q, a positive proportion of primitive characters chi in any fixed Galois orbit O (as k grows) satisfy L(1/2, chi eta1) L(1/2, chi eta2) != 0. The proof is based on a mollified second moment lower bound (Theorem 1.2) and a mollified fourth moment upper bound (Theorem 1.3), with the unconditional twisted moment asymptotic (Theorem 1.5) as the basis for the lower bound. The paper also proves unconditional second moment asymptotics for full and thin Galois orbits (Corollaries 1.6 and 1.11).","tokens_in":33022,"tokens_out":27988,"duration_ms":288944,"significance":"If the main theorem were correct, it would be a meaningful extension of simultaneous nonvanishing results to the refined family of Galois orbits of Dirichlet characters, building on the work of Khan, Milicevic, and Ngo. The use of an Euler-product mollifier and the Roth-Ridout theorem in this context is interesting, and the unconditional twisted-moment results would be of independent value. However, the proof of the central unconditional moment asymptotic (Theorem 1.5) contains what appear to be serious gaps in both the diagonal and off-diagonal contributions, and those gaps undermine the conditional main theorem as well.","major_comments":[{"comment":"The support condition for the off-diagonal sums is misstated. Lemma 2.3 gives EO[chi(n)] != 0 only if n^{p-1} ≡ 1 mod p^{k-1}. In S+(m1,m2) one has n = m1 n1 m2 n2, so setting a = m1 n1 and b = m2 n2, the nonzero orbit-average condition is (ab)^{p-1} ≡ 1 mod p^{k-1}, i.e. a^{p-1} ≡ b^{-(p-1)} mod p^{k-1}. The paper instead reduces to sums with (m1 n1)^{p-1} ≡ (m2 n2)^{p-1} mod p^{k-1}, i.e. a^{p-1} ≡ b^{p-1} mod p^{k-1}. These two conditions are not equivalent; a concrete example is p=3, alpha=3, a=2, b=14, where ab ≡ 1 mod 27 but a^2 != b^2 mod 27. Thus the reduction to D0(A,B;p^{k-1}) as defined in (6) is not justified, and the bounds in (16)–(18) do not follow. The same issue affects E−, where the analogous condition is (m1 n2 m2 n1)^{p-1} ≡ 1 mod p^{k-h} rather than the stated same-root condition. Since this step is the core of the unconditional Theorem 1.5, it is load-bearing.","section":"§3.1, proof of Theorem 1.5, equations (12)–(16)"},{"comment":"The main term M(m1,m2) is not the actual diagonal contribution. In S+(m1,m2) the summand contains the orbit-average factor EO[chi(m1 n1 m2 n2)]. For m1 n1 = m2 n2, this factor is EO[chi((m1 n1)^2)], not 1. For m1=m2=1, the diagonal sum is sum_r eta(r)/r V(pi r^2/(qX)) EO[chi(r^2)]. By Lemma 2.3, EO[chi(r^2)] != 0 only if r^{2(p-1)} ≡ 1 mod p^{k-1}; since -1 is not a (p-1)-th power modulo an odd prime power, this forces r^{p-1} ≡ 1 mod p^{k-1}, i.e. r lies in one of the p-1 Teichmuller residue classes modulo p^{k-1}. The resulting diagonal contribution is then O(log q / p^{k-1}) = o(1) as k tends to infinity, whereas the paper's M(1,1) equals L(1,eta) + o(1), which is typically of size 1. Thus the proposed main term in (14) is not merely missing a factor; it appears to be of the wrong order of magnitude. The paper does not explain why the omitted orbit-average factor can be replaced by 1.","section":"§3.1, equation (14)"},{"comment":"Because Theorem 1.5 is the unconditional basis for the mollified second moment lower bound in Theorem 1.2, and Theorem 1.1 uses Theorem 1.2 together with the fourth moment bound, the gaps in the proof of Theorem 1.5 are load-bearing for the paper's main nonvanishing result. The proof as written does not establish the claimed asymptotics, and the issues with the main term suggest that the stated main term in Theorem 1.5 may be incorrect rather than merely under-proven. Further, the off-diagonal bounds would need to be replaced by bounds for the inverse-product congruence (ab)^{p-1} ≡ 1, for which the Roth-Ridout estimates in Proposition 2.15 do not directly apply as stated.","section":"Theorems 1.2 and 1.1"}],"minor_comments":[{"comment":"The proof of Proposition 4.2 is only sketched by reference to a similar argument in [9]. Since this proposition is an essential part of the mollified fourth moment bound, a fuller proof would be helpful, though this is secondary to the issues in Section 3.","section":"Proposition 4.2"},{"comment":"The notation X* is used both for sums over integers not divisible by p and for sums over primitive characters; this is noted in the text but is a potential source of confusion.","section":"Notation"},{"comment":"There are several typos and minor errors in the introduction and remarks, e.g., 'Hecker' for 'Hecke' and 'neccesarily' for 'necessarily'; these should be corrected.","section":"Introductory remarks"}],"recommendation":"reject","confidential_remarks":"The issues in Section 3 are not cosmetic. The diagonal main term appears to be inconsistent with Lemma 2.3, and the off-diagonal reduction uses the wrong congruence. I do not see a way to repair these within the scope of the current manuscript; at minimum, the main term of Theorem 1.5 would need to be recomputed and the D0-type estimates replaced by genuinely different bounds. Given that the main theorem depends on this, rejection is warranted."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main theorem is not established as written. The problem sits in Section 3.1, in the proof of Theorem 1.5, the unconditional twisted second moment that everything else rests on. A stress-test objection I checked holds up: the off-diagonal sums are reduced to the wrong congruence. Lemma 2.3 says E_O[chi(n)] is nonzero only if n^{p-1} = 1 (mod p^{k-1}). Applied to n = m1 n1 m2 n2, the support is (m1 n1 m2 n2)^{p-1} = 1, i.e. ab = zeta (mod p^{k-1}) with a = m1 n1 and b = m2 n2. The paper instead restricts to a^{p-1} = b^{p-1}, i.e. a = zeta b. These differ: for p = 3, modulus 27, a = 2, b = 14 gives ab = 1, so (ab)^2 = 1, but a^2 = 4 and b^2 = 7. The D0(A,B;p^{k-1}) sums in (16) therefore neither contain nor bound the actual off-diagonal contribution. The same error affects E- with modulus p^{k-h}, and the thin-orbit Theorem 1.9 inherits it. This is load-bearing: Theorem 1.5 feeds Theorem 1.2, which feeds Theorem 1.1.\n\nA second, related problem: the diagonal main term (14) replaces E_O[chi(m1 n1 m2 n2)] by 1 when m1 n1 = m2 n2, but E_O[chi(t^2)] is not 1 - for the order-18 orbit modulo 27, E_O[chi(4)] = 0. So the asymptotic (15) is not the actual diagonal contribution as written. The authors would need to show that the orbit-average support condition does not change the residue.\n\nCredit where it is due: the question is natural and new - simultaneous nonvanishing in Galois orbits, where prior work only had non-simultaneous nonvanishing. The Euler product mollifier machinery in Section 4 (Theorem 1.3, Lemmas 4.3-4.4) is developed in real detail and reads coherently; the Holder step in Section 6 is fine given Theorems 1.2 and 1.3. The thin-orbit second moments and Proposition 2.15 are genuine additions. The reliance on GRH and Roth-Ridout is explicit, and the heavy use of [18] is legitimate even though the adaptation of that work is exactly where the error crept in.\n\nThe reader's CONDITIONAL verdict rested on the sketched Proposition 4.2 and the conditional input to Theorem 1.3 - real but secondary. The deeper problem is in Section 3.1.\n\nRecommendation: send it to a serious referee anyway. The problem and the machinery deserve scrutiny, and the flaw may be repairable - the correct support ab = zeta is the natural place to look. But I would not cite the main result as established.","headline":"The simultaneous nonvanishing theorem rests on an unconditional twisted-moment estimate whose off-diagonal reduction uses the wrong congruence and whose main term drops a nonzero orbit-average factor.","tokens_in":33619,"tokens_out":43367,"would_cite":false,"duration_ms":405230,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M06","11M26","11N37"],"pacs":[],"model":"deepseek-v4-flash","headline":"Under GRH, a positive proportion of characters in every Galois orbit satisfy simultaneous nonvanishing of two twisted Dirichlet L-functions.","keywords":["Dirichlet L-functions","simultaneous nonvanishing","Galois orbits","Euler product mollifier","mollified moments","Generalized Riemann Hypothesis","p-adic Roth theorem"],"falsifier":"A concrete check: verify numerically, for a fixed small odd prime p and increasing k, whether the twisted second moment average (1/|O|) Σ_{χ∈O} L(1/2,χη1)L(1/2,χη2) equals L(1,η1η2) to the error claimed in Corollary 1.6; a persistent discrepancy beyond the stated power saving would invalidate Theorem 1.5 and therefore Theorem 1.1. Alternatively, an explicit counterexample with a Galois orbit and two distinct imprimitive characters for which the nonvanishing count is o(q) would refute the main theorem.","tokens_in":32403,"feed_emoji":"📐","tokens_out":5605,"duration_ms":55582,"temperature":0.7,"pith_summary":"The paper proves that, assuming the Generalized Riemann Hypothesis, for any fixed odd prime p and any Galois orbit of primitive Dirichlet characters modulo q=p^k, a positive proportion of characters in the orbit satisfy L(1/2,χη1)L(1/2,χη2)≠0 for any two distinct imprimitive characters η1,η2. This is the first simultaneous nonvanishing result inside Galois orbits, extending an earlier single-function nonvanishing proportion. The proof works by bounding a mollified fourth moment from above via a new Euler product mollifier and a mollified second moment from below, with the lower bound relying on p-adic Diophantine approximation. If correct, the result shows the two central values are jointly nonvanishing for a positive share of the orbit as k→∞.","feed_headline":"Many Galois-orbit characters keep two L-functions nonzero","feed_subtitle":"Assuming GRH, a positive share of a Galois orbit has both twisted L-values nonzero at the central point.","key_machinery":"The Euler product mollifier M(χ)=∏_{j=0}^K E_{ℓ_j}(-P_{I_j}(χ;K)) built from prime sums over dyadic intervals (equations (30)–(31)), which molifies the central values by truncating a Dirichlet polynomial with a completely multiplicative weight a(n;K). The argument pairs it with the orbit character averages E_O[χ(n)] from the prior framework, which vanish unless $n^{{p-1}}$≡1 (mod $p^{{k-1}}$), converting off-diagonal terms into p-adic congruence sums D_0,D_1,D_2. These sums are bounded by elementary estimates plus the p-adic Roth theorem, giving the power-saving error in the twisted second moment. The fourth moment bound (Theorem 1.3) then follows from the conditional log-moment inequality of Chandee with the sharp 2v-th moment bound, and the second moment lower bound uses Granville–Soundararajan's lemma for log L(1,η) under GRH.","core_discovery":"The central claim is Theorem 1.1: under GRH, the count of χ in a fixed Galois orbit O for which L(1/2,χη1)L(1/2,χη2)≠0 is ≫ q, i.e., a positive proportion of the orbit size. The key structural discovery is that small-character twisting preserves primitivity (χη is primitive for primitive χ and imprimitive η), and that the twisted second moment over an orbit has a main term pinned down by L(1,η1η2) with power-saving error, unconditionally (Theorem 1.5). The GRH enters to dominate the fourth moment of L(1/2,χ)M(χ) by O(q) and to force the factor |L(1,η1η2)| in the mollified second moment to be bounded below.","pith_inferences":["The Euler product mollifier construction could be transferred to the family of cubic Hecke L-functions over Galois orbits, where the current nonvanishing densities are weaker; the obstruction is the analogue of the L(1,η) lower bound.","The p-adic congruence counting (Proposition 2.15) is likely to be reusable wherever orbit averages force congruences mod p^α, e.g., in higher twists or ray class characters over p-adic towers.","Replacing GRH by a sufficiently strong zero-free region below 3/4 might keep Theorem 1.5 with a smaller saving, suggesting the critical loss is only the explicit log L(1,η) expansion.","The compatibility window for thin orbits indicates a quantitative threshold in κ that could be probed numerically, possibly predicting where the second moment ceases to be dominated by the diagonal term."],"forward_implications":["For any fixed odd prime p, all Galois orbits modulo p^k get simultaneous nonvanishing with a uniform positive proportion as k→∞.","The unconditional second moment along a full orbit equals L(1,η1η2) plus a power-saving error, giving a precise average of the product of the two central values.","An unconditional weaker simultaneous nonvanishing bound of ≫ε q^{2/3−ε} follows via the Weyl subconvexity bound, so the phenomenon does not depend entirely on GRH.","Along thinner orbits (subquotients of the cyclotomic tower) the same twisted moment asymptotic holds under compatibility conditions, giving unconditional second moments there.","The sharp mollified fourth moment bound actually holds for all q→∞, not only prime powers, so the mollifying technique may apply to broader families."],"supporting_citations":[{"why":"Supplies the orbit character average lemmas and the congruence reduction framework, and the single-function nonvanishing result that this paper extends.","marker":"[18]"},{"why":"Gives the conditional upper bound for log |L(1/2,χ)| used in Lemma 4.1 to control the fourth moment.","marker":"[5]"},{"why":"Supplies the sharp GRH bound Σ|L(1/2,χ)|^{2v} ≪ q(log q)^{v^2} applied to bound the first term of the mollified fourth moment.","marker":"[42]"},{"why":"Its Lemma 8.2 expands log L(1,η) at short length under GRH, forcing the mollified second moment to be large.","marker":"[11]"},{"why":"Provides the p-adic Roth theorem that bounds the congruence sums D_2 in Proposition 2.15, giving the power-saving error.","marker":"[30]"},{"why":"Introduced the Euler product mollifier construction that the paper adapts to bound the mollified fourth moment.","marker":"[21]"}],"fun_headline_variants":["GRH gives simultaneous nonzero L-values in Galois orbits","Positive share of Galois orbit has two nonzero L-functions","Under GRH, many orbit characters avoid zeros for both twists","Simultaneous nonzero L-values for positive fraction of Galois orbit","GRH ensures a positive proportion of Galois orbit has nonzero product"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument assumes the Generalized Riemann Hypothesis for Dirichlet L-functions; without it the upper bound on the mollified fourth moment and the lower bound on the mollified second moment both break down.","fun_headline_variants_meta":{"raw":{"variants":["GRH gives simultaneous nonzero L-values in Galois orbits","Positive share of Galois orbit has two nonzero L-functions","Under GRH, many orbit characters avoid zeros for both twists","Simultaneous nonzero L-values for positive fraction of Galois orbit","GRH ensures a positive proportion of Galois orbit has nonzero product"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000607,"raw_usage":{"total_tokens":2823,"prompt_tokens":935,"completion_tokens":1888,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":551,"completion_tokens_details":{"reasoning_tokens":1801}},"tokens_in":551,"tokens_out":1888,"duration_ms":13108,"temperature":1.0,"reasoning_tokens":1801,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T19:00:08.532439+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete check: verify numerically, for a fixed small odd prime p and increasing k, whether the twisted second moment average (1/|O|) Σ_{χ∈O} L(1/2,χη1)L(1/2,χη2) equals L(1,η1η2) to the error claimed in Corollary 1.6; a persistent discrepancy beyond the stated power saving would invalidate Theorem 1.5 and therefore Theorem 1.1. Alternatively, an explicit counterexample with a Galois orbit and two distinct imprimitive characters for which the nonvanishing count is o(q) would refute the main theorem.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the orbit character average lemmas and the congruence reduction framework, and the single-function nonvanishing result that this paper extends."},{"cited_title":"Chandee, Explicit upper bounds for L-functions on the critical line , Proc","cited_arxiv_id":null,"evidence_quote":"Gives the conditional upper bound for log |L(1/2,χ)| used in Lemma 4.1 to control the fourth moment."},{"cited_title":"2, 37 pp","cited_arxiv_id":null,"evidence_quote":"Supplies the sharp GRH bound Σ|L(1/2,χ)|^{2v} ≪ q(log q)^{v^2} applied to bound the first term of the mollified fourth moment."},{"cited_title":"Granville and K","cited_arxiv_id":null,"evidence_quote":"Its Lemma 8.2 expands log L(1,η) at short length under GRH, forcing the mollified second moment to be large."},{"cited_title":"Ridout, The p-adic generalization of the Thue-Siegel-Roth theorem , Mathematika 5 (1958), 40–48","cited_arxiv_id":null,"evidence_quote":"Provides the p-adic Roth theorem that bounds the congruence sums D_2 in Proposition 2.15, giving the power-saving error."},{"cited_title":"Lester and M","cited_arxiv_id":null,"evidence_quote":"Introduced the Euler product mollifier construction that the paper adapts to bound the mollified fourth moment."}],"review_version":1}