{"id":"836ce805-6317-4dbd-b4fc-231080215f10","arxiv_id":"2607.21532","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"For primitive characters modulo a large prime, under GRH, the four twisted central L-value logarithms are asymptotically independent standard Gaussians under a weighted measure.","lead":"Under the Generalized Riemann Hypothesis, the authors prove a weighted central limit theorem for the joint distribution of four Dirichlet L-function central values over primitive characters modulo a large prime. This yields positive proportions of characters for which all four values are simultaneously large, or simultaneously nonzero and small.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Equation (3.2) is false as stated for Steinhaus X(p); the random-model bridge appears to require missing conjugates, leaving Proposition 2.4 unjustified as written.","rationale":"I read the paper as a serious contribution whose central theorem depends on (i) external results from the companion preprint [4] and (ii) an internal reduction from character sums to a Steinhaus random model. The reader's verdict identifies (i) as the weakest assumption. My stress-test found a distinct, more basic problem in (ii): equation (3.2), the stated justification for the random model, is false for the random variables as defined. A single small example (q=5, m=2, n=3) disproves it. Moreover, the subsequent random-model calculations (e.g., Lemma 4.2) are exactly the identities one obtains using E[X(a)\\overline{X(b)}], not E[X(a)X(b)], so the manuscript appears to have a systematic missing-conjugate convention. If this is only a typographical omission, the proof might be repairable by consistently inserting overlines; but as written, the central bridge (3.5) is not established. The companion dependency remains real, but the conjugation issue is more immediately load-bearing because it affects the internal logic of the paper independent of [4]. I therefore recommend UNVERDICTED pending a corrected and consistent statement of the random model, rather than ACCEPT or CONDITIONAL based on the current text. This is a good-faith reading: I am not claiming the mathematical result is false, only that the written proof does not currently support it.","tokens_in":26553,"tokens_out":17878,"duration_ms":164053,"concrete_test":"Evaluate both sides of (3.2) for q=5, m=2, n=3. The left side is 1 and the right side is 0, so the identity fails as written. To settle whether a consistent convention repairs the proof, replace every occurrence of E[X(a)X(b)] by E[X(a)\\overline{X(b)}] (and correspondingly the twisted first moment by one involving \\overline{χ(ℓ_2)}), then re-derive Lemma 3.2, Corollary 3.4, Lemma 4.2, and Lemma 4.8. If the corrected derivation reproduces the main terms of Theorem 3.1 with the condition ℓ_1 m = ℓ_2 n, the issue is a notational gap; if not, the random-model reduction is invalid.","verdict_should_be":"UNVERDICTED","load_bearing_attack":"The written reduction to the random model contains a false identity. Equation (3.2) claims that for m,n<q, (1/φ(q))∑_{χ} χ(m)χ(n) = E[X(m)X(n)], where X(p) are i.i.d. uniform on the unit circle and X(n)=∏_{p^a∥n} X(p)^a. But for Steinhaus variables, E[X(m)X(n)]=1 iff every prime exponent of m plus that of n is 0, i.e., m=n=1; otherwise it is 0. Character orthogonality gives 1 iff mn≡1 mod q. For q=5, m=2, n=3, the left side is (1/4)∑_{χ mod 5} χ(6)=1, while E[X(2)X(3)]=0. The correct random-model identity should involve a conjugate: E[X(m)\\overline{X(n)}]=δ_{m≡n} (for m,n<q). This is not a cosmetic issue: Lemma 4.2's computation E[L_p]=∏_{r,s} L_p(1+2s,χ_{j_r}χ_{j_s}) is exactly what one obtains from E[X(m)\\overline{X(n)}], not from E[X(m)X(n)]. The same missing-conjugate pattern affects (3.3), the definition of L_V(X), Lemma 4.8, and Lemma 4.9. If the intended convention is with conjugates, then every expectation and every use of Theorem 3.1 in Section 3 has to be re-derived consistently; as written, the bridge (3.5) and Corollary 3.4 are unsupported. Since Proposition 2.4 and Theorem 1.1 rely on this bridge, this is at least as load-bearing as the companion-preprint dependency identified by the reader.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves, under GRH, a weighted four-dimensional central limit theorem for log|L(1/2, χχ_j)|, j=1,...,4, as χ varies over primitive characters modulo a large prime q, with χ_1,...,χ_4 fixed even primitive characters of pairwise coprime square-free moduli D_j ≤ D. The weight F is built from four central L-values and a two-scale mollifier, so it vanishes at zeros; the normalization is sqrt((1/2) log log q). The limiting law is stated to be four independent standard Gaussians, with error O_ε((log log q)^{-1/2+ε}), under the condition D^{272} ≪ q^{11/16-1/2000}. From this the authors deduce that ≫_c q characters have all four values |L(1/2,χχ_j)| > exp(c√log log q), and a further ≫_c q characters have all four values in (0, exp(-c√log log q)). The proof reduces the weighted characteristic function of the relevant short prime sums to a random Steinhaus model, computes the model Euler product by a prime-by-prime analysis, and then uses Beurling–Selberg majorants to pass from prime sums to log|L|.","tokens_in":27132,"tokens_out":18384,"duration_ms":171661,"significance":"If the companion results quoted from [4] are valid and the random-model bridge is repaired, this is a substantial advance. The weighted approach gives a genuine two-sided joint normal law in a family where vanishing obstructs the unconditional Selberg theorem, with an explicit error term and with the variance computed directly from Σ_{p∈I0} 1/p rather than fitted. The four-dimensional statement is stronger than pairwise decorrelation, and the corollary yields a positive proportion of characters with simultaneous large or small values, going beyond earlier one-sided or pairwise results. The paper is also honest about its main external dependencies, although those dependencies are heavy.","major_comments":[{"comment":"Equation (3.2) is false as stated. For i.i.d. Steinhaus variables X(p), E[X(m)X(n)] = ∏_p E[X(p)^{a_p+b_p}], which equals 1 only when m=n=1 and is 0 otherwise. Character orthogonality gives 1 when mn≡1 mod q; for q=5, m=2, n=3, the left side of (3.2) is 1 and the right side is 0. The correct identity is E[X(m)overline{X(n)}]=δ_{m=n} for 1≤m,n<q. This is not cosmetic: Lemma 4.2 derives a=b from E[X(p)^a X(p)^b], which is true only with a conjugate; (3.3), the definition of L_V(X), Lemma 3.2, and (3.5) all use the same un-conjugated pairing. Consequently the bridge in Corollary 3.4, and with it Proposition 2.4 and Theorem 1.1, is unsupported as written. The intended argument is recognizable, but the random model must be redefined with a consistent conjugation convention and Section 4 re-derived.","section":"§3, Eq. (3.2) and (3.3)–(3.5)"},{"comment":"The same missing conjugate occurs in the proof of Lemma 2.2. For |Σ χχ_j(p)a(p)/√p|^{2k}, the expansion must include χχ_j(m) overline{χχ_j(n)} on the second factor. As printed, (2.1) has χχ_j(m)χχ_j(n), so the orthogonality condition is mn≡1 mod q, not m=n; again q=5, m=2, n=3 is a counterexample. The stated bound k!(Σ_{p∈I0}1/p)^k is the standard one and will follow after inserting the missing conjugate, but the proof as written is invalid. Since Lemma 2.2 is used for the moment estimates in Lemma 3.3 and Lemma 6.2, this needs correction.","section":"§2, Lemma 2.2, Eq. (2.1)"},{"comment":"The central claim depends on several deep results quoted from the companion preprint [4]: the twisted first-moment asymptotic (Theorem 3.1), the high mollified moment bound (1.3), the lower bound φ*_F(q)≍q, and the cross-correlation estimates used in Lemmas 4.10 and 4.11 (labeled as (7.28) and Proposition 7.2 of [4]). None of these is proved in the present manuscript. Theorem 3.1 is the engine of the random-model reduction, and (1.3) is used to control all exceptional sets. This is a load-bearing verifiability concern: the referee cannot certify the main theorem from the text alone. The authors should make the companion preprint available in full as part of the review package, or include the necessary arguments.","section":"§1, Theorem 3.1, Eq. (1.3), §4 Lemmas 4.10–4.11"}],"minor_comments":[{"comment":"The phrase 'using the fact that ∏_{p∈I0} L_{p,1,2,3,4}(1/2,X) fM_p(X) ≠ 0' is not the right justification. The random product is almost surely nonzero for trivial reasons; what is needed is that the expectation of that product is nonzero (or at least that the division is justified). Please rephrase.","section":"§4, before Lemma 4.10"},{"comment":"The condition u_j ≪ sqrt(log log q) is correct after rescaling, but it would help to state explicitly that this matches the support restriction of the Beurling–Selberg functions, where |u_j| ≤ Δ and Δ = sqrt(log log q)/log log log q.","section":"§6, after Eq. (6.1)"},{"comment":"The notation φ*_F(q) for a weighted sum is easily confused with φ^*(q) for the number of primitive characters. A different symbol, such as S_F(q), would improve readability.","section":"§2, notation"},{"comment":"The manuscript cites specific results from [4] by equation numbers ('(7.28)', 'Proposition 7.2', 'Theorem 1.5') without making the companion available in the bibliography beyond an arXiv number. Please list the exact statements or a stable reference.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The conjugation error in (3.2) seems repairable—the intended convention is clear and most subsequent computations are consistent with it—but it is pervasive enough that the current text cannot be accepted. I also recommend that the editor obtain the companion preprint [4] before the next round, since Theorem 3.1 and (1.3) are load-bearing and not proved here."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The short version: this is a real and serious paper, but Section 3 has a load-bearing conjugation error, and the proof as written cannot be trusted until that is fixed.\n\nThe new content is genuine: a four-dimensional weighted CLT for twists of four Dirichlet L-functions by primitive characters mod q, under GRH, with the log log log q decorrelation for growing conductors, plus the corollary on positive proportions of simultaneously large or small nonzero values. The overall architecture—mollified first moments, random model, Beurling–Selberg approximation—is the right one, and the pieces mostly line up. The authors are also honest about drawing heavily on their companion preprint [4] for Theorem 3.1, (1.3), and (7.28); that is a normal dependency, but it means this text alone is not self-contained.\n\nThe problem is equation (3.2). It says (1/φ(q)) Σ_χ χ(m)χ(n) = E[X(m)X(n)] for m,n<q. That is false. Character orthogonality gives 1 when mn≡1 mod q; the Steinhaus expectation is 1 only when m=n=1. Try m=2, n=3, q=5: left side is 1, right side is 0. The correct random-model identity uses a conjugate: E[X(m)\\overline{X(n)}] = δ_{m=n}. The paper's own later computations confirm this—Lemma 4.2 and Lemma 4.8 produce exactly the conjugate expectation. So the definition of L_V(X), the identity (3.3), and the bridge (3.5) are all written with the wrong character-side product. As it stands, Corollary 3.4 and Proposition 2.4 are not justified. This is not a cosmetic typo; it is the mechanism that turns the character sum into the random model.\n\nThere are smaller soft spots: Lemma 4.5 and Lemma 4.11 are sketched, and the proof of Lemma 4.11 is a bit hand-wavy. But those are secondary.\n\nIf the conjugation convention is repaired consistently—replace X(m)X(n) by X(m)\\overline{X(n)} in the relevant definitions, redo Section 3, and check the swaps—the main theorem is probably salvageable. As written, I would not cite it. I would send it to peer review anyway; a competent referee can catch exactly this issue, and the underlying program deserves careful review. The right verdict at this stage is 'revise before acceptance,' not desk rejection.","headline":"Four-twist weighted CLT with a real conjugation bug in the random-model bridge; the main theorem is plausible but Proposition 2.4 is unjustified as written.","tokens_in":27551,"tokens_out":4323,"would_cite":false,"duration_ms":39114,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M06","11M26","60F05"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that, under the Generalized Riemann Hypothesis, the four logarithms of the central values of Dirichlet L-functions twisted by a varying primitive character are asymptotically independent standard Gaussians under a weighted","keywords":["central limit theorem","central values of L-functions","simultaneous non-vanishing","Dirichlet L-functions","mollifier","random model","generalized Riemann hypothesis","twisted first moment"],"falsifier":"Compute the twisted first moment I(ℓ_1,ℓ_2) for a concrete quadruple of even primitive characters χ_1,...,χ_4 with D^272 L^96 ≪ q^{11/16-ε} and compare it with the six-term random-model expectation claimed in (3.3); any term-by-term discrepancy larger than q^{-δ} would refute the reduction and with it the theorem. A weaker but equally decisive check is the cross-correlation bound ∑_{p≤q_0} χ_j(p)conjugate(χ_k)(p)/p ≪ log log log q for j≠k with D_jD_k of size a fixed small power of q; if that bound fails, the Gaussian factors fail to be independent.","tokens_in":26464,"feed_emoji":"🎲","tokens_out":9350,"duration_ms":78927,"temperature":0.7,"pith_summary":"This paper tries to prove that, assuming the Generalized Riemann Hypothesis, the four central values of Dirichlet L-functions obtained by twisting a fixed set of four characters by a varying primitive character mod q are jointly Gaussian once each value is logarithmically scaled and the counting is done with a carefully chosen weight. The weight is built from mollified L-values, so it vanishes on characters where any of the four central values vanishes; this sidesteps the fact that log|L(1/2, χχ_j)| is undefined when the value is zero. If the theorem is right, the four normalized logarithms are asymptotically independent standard normals under this weighted measure, and the product structure gives a positive proportion of characters with all four central values simultaneously larger than exp(c sqrt(log log q)), and a positive proportion with all four nonzero and smaller than exp(-c sqrt(log log q)). That simultaneous two-sided control is new for four twists; previous methods gave only one-sided bounds. The proof reduces the character sums to a random model of iid unit-circle variables and evaluates the resulting expectations by splitting primes into small, medium, and large ranges.","feed_headline":"Four central L-values become independent Gaussians","feed_subtitle":"A weighted count makes four L-values follow one normal law, so many are all large or all small.","key_machinery":"The argument runs through a weighted measure μ_F with F = ∏_{j=1}^2 L(1/2,χχ_j)M(1/2,χχ_j) ∏_{k=3}^4 conjugate(L(1/2,χχ_k)M(1/2,χχ_k)), where M is a mollifier—a short Dirichlet polynomial engineered to approximate ∏_j L(1/2,χχ_j)^{-1}, so that the products are close to 1 for most χ and log|L| is well-approximated by -log|M|, a short prime sum. The key reduction (Corollary 3.4) replaces the character-sum average of F times an exponential of the four prime sums by the expectation of the corresponding random-model object, built from i.i.d. unit-circle random variables X(p), using the orthogonality relation (3.2) and the twisted first-moment asymptotic of the companion paper. In the random model","core_discovery":"At the top of the paper stands Theorem 1.1: assuming GRH and D^272 ≪ q^{11/16-1/2000}, for any four intervals U_1,...,U_4 the weighted proportion μ_F of primitive characters χ mod q for which log|L(1/2,χχ_j)|/sqrt(1/2 log log q) ∈ U_j for all j equals the four-dimensional Gaussian integral 1/(4π^2)∫_{U_1×...×U_4} e^{-Σ x_j^2/2} dx, up to O_ε((log log q)^{-1/2+ε}). In other words, the four normalized logarithms are asymptotically independent standard Gaussians under μ_F. The weight F is chosen so that F(χ)=0 if any of the four central values vanishes, and F≈1 for typical χ; this removes the obstruction that log|L| is undefined at zeros. The direct corollary is that for any fixed c>0, ≫_c q ch","pith_inferences":["Editorial extension: the same six-term swap structure should generalize to a k-dimensional weighted CLT for any fixed number k of twists; the four-dimensional case is the first where all six cross-swaps appear, but the mechanism—small-prime Gaussian factor plus medium/large-prime control—does not seem to depend on k being 4.","Editorial extension: because the error term O_ε((log log q)^{-1/2+ε}) matches the classical rate expected for Selberg-type theorems, one may conjecture that the true deviation from the Gaussian law is governed by the log log log q cross-correlation terms, so the stated rate may be improvable but the log log log q contamination is intrinsic to the method.","Editorial extension: if a way were found to show the weight F is asymptotically 1 on most characters (i.e., φ_F^*(q)∼φ^*(q) and F→1 in measure), the weighted CLT would upgrade to an unweighted Selberg-type CLT for the four central values, but the positivity of F is currently essential to control the undefined logarithm at zeros.","Editorial extension: the argument should adapt to simultaneous non-vanishing of more than four L-functions or to families with a twisted first-moment asymptotic of similar shape, at the combinatorial cost of a growing number of swap terms."],"forward_implications":["If Theorem 1.1 is correct, then under GRH, for every fixed c>0 there are ≫_c q primitive characters χ mod q such that |L(1/2,χχ_j)|>exp(c sqrt(log log q)) simultaneously for j=1,2,3,4.","If Theorem 1.1 is correct, then under GRH, for every fixed c>0 there are ≫_c q primitive characters such that 0<|L(1/2,χχ_j)|<exp(-c sqrt(log log q)) simultaneously for j=1,2,3,4.","The four normalized logarithms are asymptotically independent standard Gaussians under μ_F, so the limiting joint distribution factors into a product of four one-dimensional normal laws.","The weighted measure gives a genuine two-sided asymptotic, not just an upper or lower bound; this two-sided control is what supports the simultaneous large and small-value statements.","The decorrelation among distinct twists is governed by the character sums ∑_{p≤q_0} χ_j(p)conjugate(χ_k)(p)/p, whose size is log log log q rather than O(1); this is the quantitative reason the four variables become independent as q→∞."],"fun_headline_variants":["GRH: four central L-values become Gaussian and independent","Weighted CLT turns four L-values into Gaussian quadruple","GRH yields joint Gaussian law for four L-values","Four L-functions: independent Gaussians at the center","Central L-values: four independent normals under GRH"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is the correctness of the companion paper's twisted first-moment asymptotic for the fourfold product (its Theorem 3.1) together with the high mollified moment bound (1.3); the present paper imports both without reproof, and if either fails or holds on a narrower range than stated, the weighted central limit theorem collapses.","fun_headline_variants_meta":{"raw":{"variants":["GRH: four central L-values become Gaussian and independent","Weighted CLT turns four L-values into Gaussian quadruple","GRH yields joint Gaussian law for four L-values","Four L-functions: independent Gaussians at the center","Central L-values: four independent normals under GRH"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00024,"raw_usage":{"total_tokens":1317,"prompt_tokens":666,"completion_tokens":651,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":410,"completion_tokens_details":{"reasoning_tokens":570}},"tokens_in":410,"tokens_out":651,"duration_ms":6633,"temperature":1.0,"reasoning_tokens":570,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T07:16:13.234744+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the twisted first moment I(ℓ_1,ℓ_2) for a concrete quadruple of even primitive characters χ_1,...,χ_4 with D^272 L^96 ≪ q^{11/16-ε} and compare it with the six-term random-model expectation claimed in (3.3); any term-by-term discrepancy larger than q^{-δ} would refute the reduction and with it the theorem. A weaker but equally decisive check is the cross-correlation bound ∑_{p≤q_0} χ_j(p)conjugate(χ_k)(p)/p ≪ log log log q for j≠k with D_jD_k of size a fixed small power of q; if that bound fails, the Gaussian factors fail to be independent.","supporting_citations":[],"review_version":1}