{"id":"6785f460-eae2-48f8-8b78-e74f677f3bb0","arxiv_id":"1909.01338","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A zero density estimate for Dedekind zeta functions over families of Galois extensions with fixed Galois group is proved without assuming the strong Artin conjecture.","lead":"This paper proves a new estimate, unconditional on the strong Artin conjecture, for how few zeros of certain zeta functions can be near the line Re(s)=1 in families of Galois extensions with a fixed finite Galois group. It makes several consequences of a recent breakthrough by Pierce, Turnage-Butterbaugh, and Wood unconditional for many families, including prime distribution and bounds on class group torsion.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Corollary 6.2's y-range does not cover the value used in the proof of Theorem 1.1; the zero-density estimate is unproved as written.","rationale":"The reader's weakest_assumption was Proposition 4.3 (Brauer). That is a real dependency, but it is a standard external theorem and no internal contradiction is apparent from the text. By contrast, Corollary 6.2 is proved inside the paper, and the stated hypotheses do not match the proof or the subsequent application in Theorem 1.1. Since the central theorem's proof explicitly invokes Corollary 6.2, this is a load-bearing gap in the written argument. It may well be repairable, but as written the manuscript does not fully establish Theorem 1.1. This supports keeping the verdict at CONDITIONAL rather than ACCEPT, though for a more internal and concrete reason than the Brauer-citation concern.","tokens_in":23731,"tokens_out":46356,"duration_ms":463883,"concrete_test":"Independently re-derive Corollary 6.2 from Theorem 6.1. Write down the exact condition on y from the penultimate display, isolating the term Q^{5/4(m+1)}T^{m^2}/n^{1/2}, and determine whether the required lower bound is y ≫ Q^{10^8(m+1)}T^{2m^2} or y ≫ (QT)^{10^8(m+1)}. Then check whether N0=(QT)^{(2/3)·10^3m^3}, used in §7, satisfies the corrected condition for every m≥1. If it does, amend Corollary 6.2; if it does not, the proof of Theorem 1.1 must be modified, and the final exponent 10^7m^3(1−σ) must be recomputed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Corollary 6.2 states that for y ≫_m (QT)^{10^8(m+1)} the L^2-mean of prime sums is ≪ m_F(Q)(log y)^{2m^2} log u. Its proof, however, obtains the conclusion only under the condition y ≫_m Q^{10^8(m+1)}T^{2m^2}. These two conditions differ in the T-exponent: 10^8(m+1) versus 2m^2. More seriously, the proof of Theorem 1.1 applies Corollary 6.2 with y=N0=exp(M/(300η)), where M=2·10^5m^3η log(QT), so N0=(QT)^{(2/3)·10^3m^3}. For m=1 this is (QT)^{666.7}, whereas the corollary as stated requires (QT)^{2·10^8}, and the proof's condition requires Q^{2·10^8}T^2. Thus the key averaging step in §7 has no admissible y under the stated hypotheses. Since Corollary 6.2 is the bridge between the large sieve (Theorem 6.1) and the zero-detection argument, Theorem 1.1 is incomplete as written; the y-condition must be repaired, or the application must choose a larger N0 and the final exponent must be rechecked.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a zero density estimate for Dedekind zeta functions associated to Galois extensions K/Q with a fixed finite Galois group G, counting zeros of the quotient ζ_K(s)/ζ(s) in rectangles of height T and distance 1−σ from the critical line. The result is unconditional and does not assume the strong Artin conjecture, which previous work required. The proof combines Brauer's holomorphy theorem for tensor products of Artin characters, a large sieve inequality for Dedekind zeta coefficients, and the zero-detection method of Soundararajan–Thorner. Applications are given to the average error in the Chebotarev density theorem, subconvexity of Dedekind zeta functions, and ℓ-torsion in class groups.","tokens_in":23970,"tokens_out":28222,"duration_ms":243771,"significance":"If the proof is completed, the main theorem is a significant advance: it appears to be the first zero density estimate for families of Dedekind zeta functions with arbitrary fixed Galois group without assuming strong Artin, and it yields new unconditional applications to Chebotarev, subconvexity, and torsion. The paper is well-structured, uses external theorems such as Brauer's result appropriately, and does not rely circularly on the strong Artin conjecture or on the conclusions it aims to derive. The large sieve framework and the zero-detection argument are coherent in outline, and the claimed applications are clearly stated.","major_comments":[{"comment":"The y-range in Corollary 6.2 is inconsistent with both its proof and its application. The corollary as stated requires y ≫_m (QT)^{10^8(m+1)}, while the proof obtains the conclusion only under y ≫_m Q^{10^8(m+1)} T^{2m^2}. These two conditions differ in the exponent of T, and neither condition is satisfied by the value y = N0 used in §7. In the proof of Theorem 1.1, N0 = exp(M/(300η)) with M = 2·10^5 m^3 η log(QT), so N0 = (QT)^{(2·10^5 m^3)/300} = (QT)^{(2000/3)m^3}. For m = 1, this is (QT)^{666.7}, which is far smaller than the lower bound (QT)^{2·10^8} required by the stated corollary and also smaller than Q^{2·10^8} T^2 required by the proof of the corollary when T is small. The assertion in §7 that the application is valid because (2·10^5 m^3)/300 ≥ 10^8(m+1) for m ≥ 1 is numerically false. Consequently the averaging step over K in (7.8) is not justified as written. This is a load-bearing gap in the proof of Theorem 1.1: either Corollary 6.2 must be proved with a y-condition that is actually satisfied by N0 (e.g., by improving the large sieve or by choosing a smaller power of Q), or the proof must choose a larger N0 and recompute the resulting exponent in Theorem 1.1, which will likely weaken the final zero density estimate.","section":"§6, Corollary 6.2 and §7, proof of Theorem 1.1"},{"comment":"The dyadic construction of a zero-free region does not establish the claimed region. The proof applies Theorem 1.1 with T = T_j = e^j − 3 and σ = σ_j = 1 − 20δ log Q / (log Q + log(T_j + 3)). This only shows that, for non-exceptional fields, there are no zeros with β ≥ σ_j and |γ| ≤ T_j. However, the claimed zero-free region has boundary α(t) = 1 − 20δ log Q / (log Q + log(t + 3)) for a zero of height t. Since α(t) < σ_j for t < T_j, a zero with β > α(t) but β < σ_j is not excluded by the argument. Thus the conclusion that Δ_{K/Q}(t) ≥ α(t) for all 3 ≤ t ≤ exp(Q^{ε/2}) does not follow. To obtain the claimed region, one would need to apply the density estimate with σ = α(T_{j−1}) and T = T_j (or use a similar pointwise argument), and the resulting exceptional-set estimate would need to be rechecked. This affects the proof of Theorem 2.1 and Corollary 2.2, which rely on Proposition 8.7.","section":"§8, Proposition 8.7"}],"minor_comments":[{"comment":"Several exponents are difficult to read in the plain-text version, such as Q^{54(m+1)} and Q^{108(m+1)}; these should be typeset unambiguously, and the numerical comparison in §7 should be corrected to match the actual inequalities used in the proof of Corollary 6.2.","section":"Throughout, especially §6 and §7"},{"comment":"The paper relies heavily on the authors' earlier works [4], [21], and [23] for the zero-detection lemmas and the Chebotarev refinement. It would help the reader if Section 2.1 explicitly listed which results are quoted from those papers and which are new, particularly because the proof of Lemma 7.3 and the proof of Proposition 8.7 are only sketches that send the reader to [21] and [23].","section":"§2.1 and references"},{"comment":"In the contour-integral proof of Lemma 5.1, the product over ramified primes is controlled by the bound ∏_{p|D_KD_{K'}} (1+p^{-1/2})^{m^2} ≪_ε Q^ε; this is correct, but the role of the conductor bound from Proposition 4.3 in the convexity estimate could be stated more explicitly for readability.","section":"§5, Lemma 5.1"}],"recommendation":"major_revision","confidential_remarks":"The central approach is sound and the paper is likely acceptable after repair, but the gap identified in Corollary 6.2 and used in the proof of Theorem 1.1 is genuine: the application of the corollary uses a value of y that does not satisfy the stated hypotheses, and the validation inequality in §7 is numerically false. I recommend major revision rather than rejection because the flaw appears fixable by enlarging N0 or improving the large sieve, although the final exponent in Theorem 1.1 will likely need to be weakened. The authors should also re-examine the dyadic zero-free region argument in Proposition 8.7, which as written does not justify the claimed region."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this is a real theorem, but the written proof has a gap at a badly chosen exponent. The main new input is a large sieve for coefficients of ζ_K/ζ without assuming automorphy, using Brauer's holomorphy for tensor L-functions of linearly disjoint fields. That is genuinely new, and the zero-detection setup from Soundararajan–Thorner fits cleanly. The applications to Chebotarev, subconvexity, and torsion are honestly stated, with the S_n case flagged as conditional on the discriminant/biquadratic counting condition (2.7). I don't see circularity or fitted constants; the self-citations are for methodology, not for the central claim.\n\nThe soft spot is concrete. Corollary 6.2 as stated requires y ≫_m (QT)^{10^8(m+1)}; its proof, however, requires y ≫_m Q^{10^8(m+1)}T^{2m^2}. More importantly, in the proof of Theorem 1.1 they apply Corollary 6.2 with y = N0 = exp(M/(300η)), where M = 2·10^5 m^3 η log(QT), so N0 = (QT)^{(2/3)·10^3 m^3}. That is about (QT)^{6.7·10^2 m^3}, not (QT)^{10^8(m+1)}. The paper's line saying that (2·10^5)/300 ≥ 10^8(m+1) is off by roughly five orders of magnitude. Thus the averaging step that bridges the large sieve and the zero count has no admissible y under the stated hypotheses, and Theorem 1.1 is unproved as written. This is not a cosmetic typo: the natural fix, choosing M much larger, would blow up the final exponent unless other parts are reworked.\n\nDespite that, I think the proof strategy is sound and the theorem is likely true with larger exponents. The Brauer tensor input is external but standard; the Chebotarev machinery in Sections 8–10 is careful and mostly self-contained. This is the kind of paper that deserves referee time, but the authors need to repair Corollary 6.2 and the application in §7 before the main theorem is complete. I would not cite Theorem 1.1 as it stands, but I would read a revised version.\n\nRecommendation: send to peer review rather than desk reject. The result is important enough and the framework is novel enough that a serious referee should look at it, with the explicit task of checking whether the y-range can be repaired without changing the final exponent.","headline":"Genuinely new unconditional zero-density setup for nonabelian Galois families, but the written proof has a concrete exponent gap in Cor. 6.2 that leaves Thm 1.1 unproved as stated.","tokens_in":24551,"tokens_out":3050,"would_cite":false,"duration_ms":31670,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M41","11R42","11N35","11R29","11R45"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves an unconditional zero density estimate for the Dedekind zeta functions of Galois extensions with any fixed finite Galois group, removing the need for the strong Artin conjecture in recent Chebotarev and class-group…","keywords":["zero density estimates","Dedekind zeta functions","Artin L-functions","large sieve inequality","Chebotarev density theorem","class group torsion","Galois extensions","strong Artin conjecture"],"falsifier":"For a small group such as $G=S_3$ ($m=5$), enumerate all $S_3$-Galois fields with $D_K\\le Q$, compute the zeros of $\\zeta_K/\\zeta$ up to height $T$ with numerical routines, and compare $\\sum_{K\\in\\mathcal{F}(Q)}N_K(\\sigma,T)$ with $m_{\\mathcal{F}}(Q)(QT)^{107m^3(1-\\sigma)}(\\log QT)^{2m^2}$; if the ratio is unbounded for a sequence of $Q,T$, Theorem 1.1 is false. More directly, exhibiting linearly disjoint Galois $K,K'$ for which $\\zeta_{KK'}\\zeta/(\\zeta_K\\zeta_{K'})$ has a pole inside the critical strip would break Lemma 5.1 and remove the diagonal control of the large sieve.","tokens_in":23470,"feed_emoji":"🔢","tokens_out":13217,"duration_ms":128028,"temperature":0.7,"pith_summary":"The paper proves the first zero density estimate for Dedekind zeta functions of Galois extensions with a fixed finite Galois group $G$ that does not assume unproven progress toward the strong Artin conjecture. It shows that, over fields $K$ with discriminant at most $Q$, the number $N_K(\\sigma,T)$ of zeros of $\\zeta_K(s)/\\zeta(s)$ in the region $\\beta\\ge \\sigma$, $|\\gamma|\\le T$, satisfies $\\sum_{K\\in \\mathcal{F}(Q)} N_K(\\sigma,T) \\ll_m m_{\\mathcal{F}}(Q)(QT)^{107m^3(1-\\sigma)}(\\log QT)^{2m^2}$, where $m+1=|G|$ and $m_{\\mathcal{F}}(Q)$ counts how many fields in the family share a nontrivial subfield. This makes it possible to average the error term in the Chebotarev density theorem, prove subconvexity for $\\zeta_k(1/2)$, and bound $\\ell$-torsion in class groups across such families, all without the strong Artin conjecture. The price is a hypothesis on how many fields in the family share a common subfield, which is known to hold for simple Galois groups such as $A_n$.","feed_headline":"First zero density bound for Dedekind zetas without Artin conjecture","feed_subtitle":"On average over bounded-discriminant Galois fields, scarce zeros near Re(s)=1 power Chebotarev and torsion results.","key_machinery":"The engine is the large sieve inequality of Theorem 6.1. For each $K$, write $L(s,\\chi_K)=\\zeta_K(s)/\\zeta(s)=\\sum_n a_K(n)n^{-s}$. The coefficients are encoded through Schur polynomials at the local roots, and the Cauchy identity rewrites the correlation sum $\\sum_{(n,D_KD_{K'})=1}a_{K\\times K'}(n)\\phi(T\\log n\\,/\\,x)$ as a contour integral of the Artin $L$-function $L(s,\\chi_K\\otimes\\chi_{K'})$. When $K\\cap K'=\\mathbb{Q}$, the holomorphy and conductor bound of Proposition 4.3 let the contour shift to $\\mathrm{Re}(s)=1/2$, yielding the saving $\\sqrt{x}\\,Q^{m/2+\\varepsilon}T^{m^2}$; when the fields intersect, a trivial divisor-counting bound loses only the factor $m_{\\mathcal{F}}(Q)$. A standard mean-value identity from the large sieve literature converts the resulting short-interval bound into an $L^2$-averaged bound over primes, and the zero-detection lemmas of Section 7 convert that bound into the zero count $N_K(\\sigma,T)$.","core_discovery":"The central claim is Theorem 1.1: for any nontrivial finite group $G$ of order $m+1$, the sum over Galois extensions $K/\\mathbb{Q}$ with $\\mathrm{Gal}(K/\\mathbb{Q})\\cong G$ and $D_K\\le Q$ of the zero count $N_K(\\sigma,T)$ for $\\zeta_K(s)/\\zeta(s)$ is $\\ll_m m_{\\mathcal{F}}(Q)(QT)^{107m^3(1-\\sigma)}(\\log QT)^{2m^2}$ for $1/2\\le \\sigma\\le 1$. The exponent $107m^3(1-\\sigma)$ is the essential feature: near $\\sigma=1$, very few zeros occur, which is precisely what average prime counting requires. The novelty is that the averaging does not require each $\\zeta_K/\\zeta$ to be automorphic; instead, the proof uses a large sieve inequality for the Dirichlet coefficients of these Artin $L$-functions, whose diagonal part is controlled by the holomorphy, quoted as Proposition 4.3, of the tensor-product Artin $L$-function $L(s,\\chi_K\\otimes\\chi_{K'})$ for linearly disjoint fields. From the zero density theorem, the paper derives a positive level of distribution for Chebotarev primes, subconvexity of Dedekind zeta functions, and nontrivial class-group torsion bounds, all under the single hypothesis $m_{\\mathcal{F}}(Q)\\ll_{m,\\varepsilon}Q^{-2\\varepsilon}|\\mathcal{F}(Q)|$.","pith_inferences":["Editorial inference beyond the paper: the same large sieve should apply to any family of Artin representations for which the tensor-product $L$-functions are known to be entire from a source other than automorphy; the method is not logically tied to the specific quotient $\\zeta_K/\\zeta$.","Editorial inference beyond the paper: the exponent $107m^3$ is presumably far from sharp; any sharper conductor estimate for $L(s,\\chi_K\\otimes\\chi_{K'})$ would lower the exponent and improve the level-of-distribution exponent $\\delta=\\varepsilon/(109m^3)$.","Editorial inference beyond the paper: condition (2.6) turns into a discriminant multiplicity problem for groups with a unique nontrivial normal subgroup; the paper's treatment of $S_n$ shows that resolving enough of the discriminant multiplicity conjecture would make the $S_n$ case unconditional.","Editorial inference beyond the paper: Theorem 1.1 also provides, for most fields in such families, a zero-free region of width roughly $1/\\log Q$ near the edge of the critical strip, which is the natural unconditional proxy for GRH in these families and could serve as the input to single-field effective Chebotarev estimates."],"forward_implications":["With condition (2.6), the averaged Chebotarev error over all conjugacy classes is $O(x/(\\log x)^A)$ for $\\log Q$ up to $x^{\\varepsilon/(109m^3)}$, a positive level of distribution without GRH or the strong Artin conjecture.","For the completely split conjugacy class, the average error is $O(x^{1-\\delta})$, a stronger saving on the primes that split completely.","All but $O(Q^{-\\varepsilon}|\\mathcal{F}(Q)|)$ fields have $|\\zeta_k(1/2)|\\ll D_k^{1/4-\\delta/109}$ for every subfield $k$ whose Galois closure is in the family, giving subconvexity on average.","All but $O(Q^{-\\varepsilon}|\\mathcal{F}(Q)|)$ fields have $|\\mathrm{Cl}_k[\\ell]|\\ll D_k^{1/2-1/(2\\ell([k:Q]-1))+\\eta}$, a nontrivial bound for $\\ell$-torsion in class groups.","For $G=A_n$, $n\\ge 5$, the hypotheses are unconditional, giving the first unconditional instances of these three conclusions for infinitely many unsolvable extensions of arbitrarily large degree."],"supporting_citations":[{"why":"Supplies Proposition 4.3, the holomorphy of the tensor-product Artin L-function for linearly disjoint fields; this is what makes the diagonal terms in the large sieve (Lemma 5.1) small.","marker":"[3]"},{"why":"Provides the Hadamard-product zero-detection lemmas and the central L-value bound (Lemmas 7.1, 7.3-7.5) that convert the large sieve into the zero density estimate.","marker":"[21]"},{"why":"Provides the mean-value identity used in Corollary 6.2 to pass from short-interval coefficient sums to the L^2 average over primes and t in the zero count.","marker":"[8]"},{"why":"Contains the prior effective Chebotarev and torsion results that this paper strengthens, and the field-counting results used to verify condition (2.6) for alternating and symmetric groups.","marker":"[20]"},{"why":"Supplies the field-counting bound #F(Q) << Q^{5/2(m+1)} used to close the large sieve inequality and the reflection-principle lemma used for the class-group torsion application.","marker":"[7]"},{"why":"Provides the template zero density estimate for automorphic families and the weighted-prime Chebotarev machinery whose refinements appear as Propositions 8.2 and 8.4.","marker":"[4]"},{"why":"Supplies the explicit weight function and the auxiliary Chebotarev lemmas (Lemma 10.1 and surrounding estimates) used in Section 10 to prove the averaged prime counting theorems.","marker":"[23]"}],"fun_headline_variants":["Zero density for Dedekind zetas, no strong Artin","Bypassing Artin: zero density for Dedekind zetas","First zero density estimate for Dedekind zeta families","Zero density for Dedekind zetas without strong Artin conjecture"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing external input is the theorem, quoted as Proposition 4.3, that for linearly disjoint Galois extensions $K,K'/\\mathbb{Q}$ the tensor-product Artin $L$-function $L(s,\\chi_K\\otimes\\chi_{K'})$ is entire with conductor dividing $D_K^{m'}D_{K'}^m$; if that holomorphy were unavailable, the diagonal estimate in the large sieve (Lemma 5.1) and hence the whole Theorem 1.1 would collapse.","fun_headline_variants_meta":{"raw":{"variants":["Zero density for Dedekind zetas, no strong Artin","Bypassing Artin: zero density for Dedekind zetas","First zero density estimate for Dedekind zeta families","Zero density for Dedekind zetas without strong Artin conjecture"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001078,"raw_usage":{"total_tokens":4520,"prompt_tokens":967,"completion_tokens":3553,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":583,"completion_tokens_details":{"reasoning_tokens":3479}},"tokens_in":583,"tokens_out":3553,"duration_ms":24040,"temperature":1.0,"reasoning_tokens":3479,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T05:22:28.548720+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a small group such as $G=S_3$ ($m=5$), enumerate all $S_3$-Galois fields with $D_K\\le Q$, compute the zeros of $\\zeta_K/\\zeta$ up to height $T$ with numerical routines, and compare $\\sum_{K\\in\\mathcal{F}(Q)}N_K(\\sigma,T)$ with $m_{\\mathcal{F}}(Q)(QT)^{107m^3(1-\\sigma)}(\\log QT)^{2m^2}$; if the ratio is unbounded for a sequence of $Q,T$, Theorem 1.1 is false. More directly, exhibiting linearly disjoint Galois $K,K'$ for which $\\zeta_{KK'}\\zeta/(\\zeta_K\\zeta_{K'})$ has a pole inside the critical strip would break Lemma 5.1 and remove the diagonal control of the large sieve.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies Proposition 4.3, the holomorphy of the tensor-product Artin L-function for linearly disjoint fields; this is what makes the diagonal terms in the large sieve (Lemma 5.1) small."},{"cited_title":"Soundararajan and J","cited_arxiv_id":null,"evidence_quote":"Provides the Hadamard-product zero-detection lemmas and the central L-value bound (Lemmas 7.1, 7.3-7.5) that convert the large sieve into the zero density estimate."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the mean-value identity used in Corollary 6.2 to pass from short-interval coefficient sums to the L^2 average over primes and t in the zero count."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Contains the prior effective Chebotarev and torsion results that this paper strengthens, and the field-counting results used to verify condition (2.6) for alternating and symmetric groups."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the field-counting bound #F(Q) << Q^{5/2(m+1)} used to close the large sieve inequality and the reflection-principle lemma used for the class-group torsion application."},{"cited_title":"Zeros of Rankin-Selberg $L$-functions at the edge of the critical strip","cited_arxiv_id":"1804.06402","evidence_quote":"Provides the template zero density estimate for automorphic families and the weighted-prime Chebotarev machinery whose refinements appear as Propositions 8.2 and 8.4."},{"cited_title":"Thorner and A","cited_arxiv_id":null,"evidence_quote":"Supplies the explicit weight function and the auxiliary Chebotarev lemmas (Lemma 10.1 and surrounding estimates) used in Section 10 to prove the averaged prime counting theorems."}],"review_version":1}