{"id":"e53175b3-5758-4d1d-a516-8a8c1d27f223","arxiv_id":"2508.20814","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"An explicit Tauberian theorem using twisted moment bounds instead of pointwise bounds yields square-root-saving error terms for counting C_n-extensions of Q for n = 3, 4, 8, 16, and 2p.","lead":"This paper proves a Tauberian theorem that uses only average-size bounds on a Dirichlet series, not pointwise bounds, and applies it to count abelian number fields with square-root-saving error terms for cyclic groups of orders 3, 4, 8, 16, and twice an odd prime.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 5.2's higher-moment generalization is unproved and the stated lemma is false for K=Q, m=1; the corollaries for C4/C6/C8/C16/C2p depend on it.","rationale":"Theorem 2.1 was reviewed independently: the contour-shifting and unsmoothing arguments are written out, no pointwise bound is invoked, and the optimization over T is consistent. The main theorem is therefore likely correct. The applications, however, all pass through Lemma 5.2 and Lemma 5.3 to verify the averaged-input condition (1.1). Lemma 5.2 is the gatekeeper for the positive-power factors, and its m>1 cases are not proven. Moreover, the stated lemma is false for K=Q, m=1 in an unused regime (σ in (0,1/2)), which shows the blanket statement is not a safe quote from [CN63] and weakens the credibility of the 'proven similarly' assertion for m>1. Lemma 5.3's negative-exponent applications are a further citation gap, but the primary blocker is Lemma 5.2. Thus the Reader's CONDITIONAL verdict is appropriate: the Tauberian machinery is accepted, while Corollary 1.3 should be regarded as conditional on fillable moment estimates.","tokens_in":24433,"tokens_out":31878,"duration_ms":294068,"concrete_test":"For each pair (m,K) actually used in the corollaries—(2,Q), (2,Q(i)), (2,Q(ζ8)), (2,Q(ζ16)), (3,Q(ζ6))—independently derive the approximate functional equation for ζ_K(s)^m following [CN63, Eq. (65)] and compute the resulting moment bound at σ=1−1/(m[K:Q]). Accept Lemma 5.2 only if each bound is at most C T (log T)^{m^2[K:Q]}. As a minimal control, verify that the m=2, K=Q case at σ=1/2 reproduces the classical T(log T)^4; any mismatch, or any AFE that cannot be established for powers, would invalidate the corollary inputs.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The applications in Corollary 1.3 reduce to verifying the twisted-moment hypothesis (1.1) via Hölder and the moment bounds of §5.2. The decisive input is Lemma 5.2, stated for every integer m≥0, but not proven there. The author's own caveat after Lemma 5.2 admits that [CN63, Thm 4] only handles m=1, and the higher-m cases are asserted 'proven similarly' with no approximate functional equation for ζ_K(s)^m supplied. The gap is not merely expository: as stated, Lemma 5.2 is false for K=Q, m=1, where the threshold σ>1−1/[K:Q]=0 would imply ∫_0^T |ζ(σ+it)|^2 dt << T for every σ>0, contradicted by the functional-equation bound ∫_0^T |ζ(σ+it)|^2 dt ≍ T^{2−2σ} (up to log factors) for 0<σ<1/2. So the statement cannot be a faithful restatement of [CN63], and the m≥2 bounds used in the C4/C6/C8/C16/C2p proofs (m=2 for ζ and several cyclotomic ζ's; m=3 for ζ_{Q(ζ6)}) rest on an unsupported assertion. If the true bounds at σ=1−1/(m[K:Q]) are larger than T(log T)^{m^2[K:Q]}, the error terms of Corollary 1.3 do not follow from Theorem 2.1, even though Theorem 2.1 itself is sound.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves an explicit Tauberian theorem (Theorem 2.1, with the less explicit form Theorem 1.1) that bounds the difference between a summatory function and its residue expansion using only averaged/twisted moment bounds of the form ∫_T^{2T} L(σ+it)Z^{it} dt, rather than pointwise vertical bounds |L(σ+it)| ≪ (1+|t|)^ξ. The proof follows Landau's finite-differencing method: smoothing, contour shift with an auxiliary average, then unsmoothing. Sections 3–4 are detailed and internally coherent. The paper then applies the theorem to counting C_n-extensions of Q for n = 3, 4, 8, 16, and 2p, claiming unconditional square-root-saving error terms. The applications reduce the twisted-moment hypothesis to Hölder-type bounds for integral moments of Dedekind zeta functions, via Lemmas 5.2 and 5.3.","tokens_in":24784,"tokens_out":21602,"duration_ms":231977,"significance":"The Tauberian theorem itself is a genuine and useful contribution: it removes the pointwise-growth input that is standard in this circle of methods, tracks parameter dependence explicitly, and the contour argument with the auxiliary average is inventive. The paper also correctly identifies the connection to moments of L-functions. If the moment inputs were fully justified, the corollaries would be a significant improvement for the listed cyclic groups. However, the application section rests on moment assertions that are not established as stated, including a visibly false special case. The lasting value of the paper is therefore the general Tauberian framework; the numerical corollaries need correction or re-proof before they can be accepted as unconditional.","major_comments":[{"comment":"The lemma is false as stated. For K=Q and m=1 it asserts I_1(σ, ζ; T) = ∫_0^T |ζ(σ+it)|^2 dt ≪ T for every σ > 0. But for 0 < σ < 1/2 the functional equation gives |ζ(σ+it)|^2 ≍ t^{1−2σ}|ζ(1−σ+it)|^2, and since 1−σ > 1/2, the integral grows like T^{2−2σ} up to log factors, not O(T). The author's own caveat states that [CN63, Thm 4] is only for m=1; but even the m=1 statement is not a faithful restatement. Thus the assertion that the m≥2 cases are 'proven similarly' is unsupported, and the higher-m bounds used in the C4, C6, C8, C16, and C2p proofs are not established. This is load-bearing: those corollaries' error exponents depend directly on these moment bounds. A correct proof or reference for the needed higher moments, and a corrected threshold, are required.","section":"§5.2, Lemma 5.3"},{"comment":"Lemma 5.3 asserts I_z(σ, ζ_K; T) ≪ T for every complex z and every σ ≥ 1, but I_z is not defined for negative z, and the applications use I_{-2} and I_{-4} at the boundary argument 4σ = 1 (e.g., in the C3, C4, C6, and C8 proofs). Negative powers require lower bounds on |ζ_K(σ+it)|, not just upper bounds. The quoted [BIR93, Theorem 3] and the remark about Brauer induction concern positive powers/Artin L-functions; they do not cover the reciprocal of ζ_K. Even for z=-1, K=Q, standard lower bounds give only |ζ(1+it)|^{-1} ≪ log t, which would produce a log factor in the integral. The exact β-values in Corollary 1.3 are therefore not justified by the text, even though the main power saving might survive with a larger logarithm exponent.","section":"§5.2, Lemma 5.3"},{"comment":"The C16 proof states that the 'first four factors' are bounded using Lemma 5.2. But one of those factors is ∫ |ζ_{Q(ζ16)}(15σ+15it)|^{-4} dt, which has a negative exponent and is outside the range m ≥ 0 of Lemma 5.2. It must rely on Lemma 5.3 (or another negative-moment input). This misattribution, together with the issues in Lemma 5.3, means the C16 error term as printed is not proven. The same pattern appears in the C2p proof. This is a local but consequential error in the application section.","section":"§5.3, proof for C16"}],"minor_comments":[{"comment":"The integrals defining the polynomial bound for twisted moments are printed as ∫_T^{T0} L(σ+it,N) Z^{it} dt. With T0 fixed, this is empty for T > T0; presumably the intended range is ∫_{T0}^{T} or ∫_T^{2T}. The proofs in Section 4 consistently use intervals [T0,T] or [T,2T], so this is a typographical/notation issue, but it should be fixed.","section":"§2.1"},{"comment":"The line 'asserting the number of C8-étale algebras is given by' should read 'C16-étale algebras' in the sentence just after applying Theorem 1.1.","section":"§5.3, C16 proof"},{"comment":"The title in the header contains 'A VERAGED'; presumably 'AVERAGED'. There is also a recurring typo 'Dirichelt' for 'Dirichlet' in Section 5. These are harmless but should be cleaned up.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The core Tauberian theorem seems sound and interesting, but the advertised number-field corollaries are not established because the moment lemmas in Section 5.2 are either false as stated or not proved for the ranges used. I would recommend major revision: the author should either supply a correct proof of the needed higher moments (and the negative-moment bounds), or weaken/revise the corollaries accordingly. The false K=Q, m=1 case suggests that the author should re-read [CN63] carefully; the stated threshold is not the correct threshold for the second moment of ζ."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The core of this paper is a Tauberian theorem whose error term depends only on twisted-moment bounds, not pointwise bounds. That is genuinely new, and the proof via finite differencing with an auxiliary average is written out carefully. The explicit tracking of constants makes it a usable tool for arithmetic counting problems. I believe Theorem 2.1 is correct as proven.\n\nThe problem is the applications. Lemma 5.2, as stated, is false. For K=Q, m=1 it claims ∫|ζ(σ+it)|² dt ≪ T for every σ>0, but for 0<σ<1/2 the integral is of order T^{2−2σ}, which is larger than T. The author notes that [CN63] only proves the m=1 case and asserts the higher moments are “proven similarly,” but no approximate functional equation for ζ_K(s)^m is supplied. That is a real gap, not just a missing reference.\n\nThat said, the specific applications use the lemma at arguments ≥1/2, where the bounds are actually true: at σ=1/2 you get T log^{m²} T, and for σ>1/2 you get O(T). So Corollary 1.3 may well survive with a corrected lemma, and the error terms would even improve in some cases. But as written, the proof does not establish the stated bounds. The log exponents in Lemma 5.2 look wrong for general [K:Q], and the higher-m moment bounds are unsupported. The C3 case, for instance, uses m=1 for Q(ζ3) at real part 1/2; the true log power there is 1, not 2, so the paper's claimed error term is weaker than what a correct argument would give.\n\nThe sieve from étale algebras to fields is summarized rather than shown, but it follows an existing template, and the explicit constant computation in Proposition 5.5 is a nice addition. The author is candid about the CN63 caveat, which I appreciate.\n\nI would send this to referees, with a clear request to fix Lemma 5.2 before publication. The Tauberian theorem itself is solid and publishable; the applications need revision to either prove the needed moment bounds or state them from a correct source.","headline":"The averaged-input Tauberian theorem is a real new tool and its proof looks sound; the applications, however, rest on a moment lemma that is false as stated, so the corollaries are conditional pending a fix.","tokens_in":25317,"tokens_out":15439,"would_cite":true,"duration_ms":137173,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M45","11R45","11M06"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves a Tauberian theorem whose error term is controlled by twisted-moment averages rather than by pointwise vertical bounds of the Dirichlet series, and uses it to obtain unconditional square-root-saving error bounds for the num","keywords":["Tauberian theorem","Dirichlet series","twisted moments","abelian number fields","discriminant counting","cyclic extensions","Dedekind zeta moments","square-root savings"],"falsifier":"Evaluate I_3(5/6, ζ_{Q(ζ_6)}; T) = ∫_0^T |ζ_{Q(ζ_6)}(5/6+it)|^6 dt for growing T: Lemma 5.2 predicts growth at most T (log T)^18. If the measured log-power or growth exponent is larger, the moment input used for C6 fails and the unconditional error bound in Corollary 1.3 for C6 does not follow, even though the conditional theorem remains valid.","tokens_in":1755,"feed_emoji":"🧮","tokens_out":1931,"duration_ms":104868,"temperature":0.7,"pith_summary":"This paper establishes a new Tauberian theorem for Dirichlet series: to pass from coefficient averages to an asymptotic for a summatory function, one no longer needs a pointwise bound on |L(σ+it)| along vertical lines. Instead, a bound on twisted-moment averages of L is enough, and the resulting error term is tracked explicitly in terms of the average-growth exponent, the pole data, and contour values. The theorem is stated self-containedly so that other counting problems can use it directly. As a demonstration, the paper proves unconditional square-root-saving error bounds for counting cyclic extensions of Q by discriminant for C3, C4, C6, C8, C16, and C2p—families where such error terms were previously available only conditionally or not at all.","feed_headline":"Six cyclic field counts get square-root-saving errors","feed_subtitle":"A Tauberian theorem using averaged twisted moments replaces pointwise growth bounds, proving sharper asymptotics.","key_machinery":"The central object is the twisted-moment integral of the Mellin transform along a vertical line, together with an auxiliary average of the contour shift in Perron's formula. The method smooths N by repeated integration, represents the smooth sum as a contour integral, shifts the contour in a u-dependent way while averaging over u, and then unsmooths using finite-difference operators. Each shifted contour piece is bounded by the twisted-moment hypothesis and integration by parts, not by pointwise estimates on L. A single floating parameter T controls the trade-off between the residue terms, the averaged integral on the leftmost line, and the outer contour pieces, and is then optimized.","core_discovery":"The central claim is Theorem 2.1: given a nondecreasing counting function N whose Mellin transform L(s,N) is meromorphic up to Re(s)=σ_a−δ and whose twisted averages satisfy a polynomial bound, the difference between N(X) and the sum of residues S_0(X) is bounded by explicit expressions involving only the moment exponents η, β, the constant Q, the residues, and the size of L(s)/s on a fixed contour. No pointwise bound of the form |L(σ+it)| ≪ (1+|t|)^ξ is used anywhere. Optimizing a floating contour parameter T yields the error X^{σ_a−δ}/max{η,1} (log X)^θ. For the applications, the twisted-moment hypothesis is verified by Hölder's inequality and integral-moment bounds for Dedekind zeta funct","pith_inferences":["The same template should convert any improved bound for twisted moments of L-functions into improved counting error bounds, so conjectural moment results would have immediate arithmetic consequences.","The restriction to C3, C4, C6, C8, C16, and C2p appears to reflect the availability of Dedekind-zeta moment bounds rather than a limitation of the theorem; verifying additional moment bounds would likely yield square-root savings for other abelian groups.","Because the error terms are explicit, one could first evaluate R_N(X) for a concrete Dirichlet series and then optimize T numerically, producing fully explicit asymptotic inequalities rather than order-of-magnitude statements.","The same averaged-input contour argument could be applied to lattice-point and divisor-sum problems whose Mellin transforms have known moment bounds but no usable pointwise vertical bounds."],"forward_implications":["If the twisted-moment hypothesis holds, Tauberian error terms depend only on average growth of L, not on pointwise growth, as summarized in Theorem 1.1 and made explicit in Theorem 2.1.","Known pointwise bounds imply the new form by taking η=ξ+1+ε, recovering earlier errors of the shape O(X^{σ_a−δ}/(ξ+1+ε)).","Existing integral-moment bounds for L-functions can be plugged in 'out of the box' to prove new asymptotic expansions, for example in counting ideals in abelian number fields.","The counts of C_n-extensions of Q of bounded discriminant have unconditional square-root-saving error terms for n=3,4,6,8,16 and n=2p for odd primes p.","The method separates the contour-shifting part from the use of a functional equation, so future improvements in moment bounds can be converted directly into better counting errors."],"supporting_citations":[{"why":"Supplies the approximate-functional-equation moment bound for Dedekind zeta functions (Lemma 5.2) used to verify the averaged-input hypothesis.","marker":"[CN63]"},{"why":"Supplies the sharp moment bound at the edge of the critical strip (Lemma 5.3) used for negative-moment factors such as ζ(·)^{-1}.","marker":"[BIR93]"},{"why":"Supplies the pointwise inverse-zeta bound (Lemma 5.4) needed for the C16 and C2p cases where Hölder plus Lemma 5.3 is insufficient.","marker":"[Ten15]"},{"why":"Provides the meromorphic continuation of the generating Dirichlet series for abelian extensions as well as previous power-saving results used in the sieve for C_p.","marker":"[Alb24b]"},{"why":"Gives the Euler-product shape for the generating series of abelian extensions over Q that Proposition 5.1 factors.","marker":"[Wri89]"},{"why":"Supplies the finite-differencing contour method that the proof adapts to averaged inputs.","marker":"[Lan15]"},{"why":"Provides a smoothed contour-shifting lemma and the pointwise-bound Tauberian theorem used for comparison with the new error terms.","marker":"[PTBZ25]"},{"why":"Supplies the unsmoothing inequality used when the counting function is not monotone but is bounded by a monotone function.","marker":"[Rou11]"}],"fun_headline_variants":["Cyclic field counts get sharper via averaged-moment Tauberian theorem","Averaged L-function moments yield square-root errors for cyclic fields","Tauberian theorem with averaged inputs tightens number-field counting","Square-root saving errors for C_n-extensions via averaged moments","Averaged twisted moments improve asymptotic counting for abelian fields"],"cache_read_input_tokens":26880,"weakest_assumption_plain":"The load-bearing premise is that the twisted-moment bounds hold for the generating series used, which the paper verifies via Hölder and the Dedekind-zeta moment bound Lemma 5.2; that lemma is cited for the first moment only and assumed to extend to all higher integer moments.","fun_headline_variants_meta":{"raw":{"variants":["Cyclic field counts get sharper via averaged-moment Tauberian theorem","Averaged L-function moments yield square-root errors for cyclic fields","Tauberian theorem with averaged inputs tightens number-field counting","Square-root saving errors for C_n-extensions via averaged moments","Averaged twisted moments improve asymptotic counting for abelian fields"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000743,"raw_usage":{"total_tokens":3163,"prompt_tokens":771,"completion_tokens":2392,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":515,"completion_tokens_details":{"reasoning_tokens":2302}},"tokens_in":515,"tokens_out":2392,"duration_ms":20469,"temperature":1.0,"reasoning_tokens":2302,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T14:47:54.224695+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate I_3(5/6, ζ_{Q(ζ_6)}; T) = ∫_0^T |ζ_{Q(ζ_6)}(5/6+it)|^6 dt for growing T: Lemma 5.2 predicts growth at most T (log T)^18. If the measured log-power or growth exponent is larger, the moment input used for C6 fails and the unconditional error bound in Corollary 1.3 for C6 does not follow, even though the conditional theorem remains valid.","supporting_citations":[],"review_version":1}