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REVIEW 3 major objections 4 minor 20 references

Simultaneous non-vanishing of Dirichlet $L$--functions, II: Weighted central limit theorem

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Pith's one-line read 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

desk verdict 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. read the letter →

arxiv 2607.21532 v1 pith:UPYVQRF4 submitted 2026-07-23 math.NT

Hung M. Bui , Alexandra Florea , Micah B. Milinovich This is my paper · ORCID
classification math.NT MSC 11M0611M2660F05
keywords centrallimittheoremvaluesofL-functionssimultaneousnon-vanishingDirichletmollifierrandommodelgeneralizedRiemannhypothesistwistedfirstmoment
topics The Riemann Hypothesis
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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

What would settle it

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.

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Extended reading notes

Core claim

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

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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→∞.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

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|.

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 (3)
  1. [§3, Eq. (3.2) and (3.3)–(3.5)] 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.
  2. [§2, Lemma 2.2, Eq. (2.1)] 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.
  3. [§1, Theorem 3.1, Eq. (1.3), §4 Lemmas 4.10–4.11] 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.
minor comments (4)
  1. [§4, before Lemma 4.10] 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.
  2. [§6, after Eq. (6.1)] 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.
  3. [§2, notation] 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.
  4. [References] 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.

Circularity Check

3 steps flagged · score 4.0 of 10

No fitted-input circularity; score reflects load-bearing self-citation to the same-authors companion [4] for the twisted first moment, moment bounds, and off-diagonal cancellations.

  1. self citation load bearing [Section 3, Theorem 3.1 (reproducing [4, Theorem 1.3]); used in (3.3)-(3.5)]
    "A crucial input in the proof is the asymptotic evaluation, established in [4], of the twisted first moment of the fourfold product, 1/φ*(q) Σ*_χ ∏_{j=1}^2 L(1/2,χχ_j) ∏_{k=3}^4 L(1/2,χχ_k)χ(ℓ_1)χ(ℓ_2), valid uniformly in the range D^{272}L^{96} ≪ q^{11/16-ε}."

    The random-model bridge (3.3) is exactly this theorem: I(ℓ1,ℓ2)=E[L_V(X)X(ℓ1)X(ℓ2)] plus an error, with L_V built from the six term-types of Theorem 3.1. The paper does not prove Theorem 3.1; it imports it from [4], a preprint by the same three authors. Thus Proposition 2.4 and Theorem 1.1 inherit the companion's twisted first moment without independent verification in this text. This is a load-bearing self-citation rather than a derivation internal to the paper.

  2. self citation load bearing [Section 1, equations (1.3)-(1.4); used in Lemmas 3.2-3.3, Corollary 3.4, and Section 6]
    "It was proved in [4, Theorem 1.4] that \sum*_{χ(mod q)} |L(1/2,χχ_j)M(1/2,χχ_j)|^k ≪_k q (1.3) for every k∈N, and that φ*_F(q)≍q (1.4)."

    These bounds control the exceptional set and the normalization of the weighted measure. Lemma 3.3 and Corollary 3.4 use (1.3) to discard χ outside the good set S, and the normalization (1.4) is used in the final step of Proposition 2.4 and in Corollary 1.2. Both are cited from the same authors' companion [4] and are not proved here, so the positive-proportion conclusion depends on a same-author self-citation for the high mollified moments.

1 more flagged steps
  1. self citation load bearing [Section 5.1, proof of Proposition 4.1, after equation (5.1)]
    "From Proposition 7.2 in [4] (see the equation following equation (7.30)), note that for (j1,...,j4) not equal to the diagonal term (i.e., j_k = k for all k), E[L_{j1,j2,j3,j4}(1/2,X)M(X)] ≪ (log q)^{-3}."

    This off-diagonal estimate is what lets the proof discard all five swapped terms and retain only the diagonal contribution in the random model, yielding the four-dimensional Gaussian factorization of Lemma 4.10 and Proposition 4.1. The result is cited from [4] by the same authors and is not derived in the present paper. If this bound failed, the claimed asymptotic independence of the four twists would not follow from the written argument.

full rationale

The paper's central claim is not a fitted-input prediction. The Gaussian in Proposition 2.4 is computed from P(χχ_j)=ℜΣ_{p∈I0}χχ_j(p)a(p;K)/√p; the variance is obtained from Σ_{p∈I0}1/(4p)+O(log log log q), with no parameter tuned to the target normal law. The Fourier-inversion and Beurling-Selberg steps in Section 6 are self-contained. So patterns 1, 2, and 6 are absent. The circularity burden is self-citation: the random-model reduction and Gaussian factorization are supported by the same-authors' companion [4] in several load-bearing places. Theorem 3.1 is imported, not proved; (1.3)-(1.4) are imported; the off-diagonal bound Proposition 7.2 is imported; the χ_jχ_k cross-correlation bound (7.28) is also imported. If any of these companion results fails, Proposition 2.4 and Theorem 1.1 collapse. Because the companion is an unpublished same-team preprint, this is self-citation load-bearing rather than external evidence. However, there is substantial independent content here (Euler-product random model computations, Lemmas 4.8-4.11, the Beurling-Selberg approximation), so the appropriate score is 4 rather than 6-8. I also flag, as a correctness risk outside the circularity score, equation (3.2): as written, for Steinhaus X, E[X(m)X(n)] does not generally equal (1/φ(q))Σ_χ χ(m)χ(n); the subsequent computations appear to need E[X(m)\overline{X(n)}]. This would make the written bridge (3.2)/(3.3)/(3.5) unsupported, but that is an error, not a self-referential reduction.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new physical or structural entities. The free parameters are hand-tuned constants in the mollifier and smoothing procedures. The main axioms are GRH and the authors' own companion results from [4], which are not independently established in this paper.

free parameters (5)
  • λ = unique solution to e^{-λ} = λ + λ^2/2
    Chosen by hand to shape the mollifier a(p;k); not fitted to data.
  • β_k = e^k/(log log q)^5, with β_K < 10^{-25} = β_k = e^k/(log log q)^5, β_K < 10^{-25}
    Partitions the primes into intervals I_k; the exponents are tuned to balance errors in the mollifier and random-model estimates.
  • ℓ_k = ℓ_k = 2⌊(1/21^{1/4})(1/(8β_k))^{3/4}⌋
    Truncation lengths for the mollifier; chosen ad hoc to control high moments.
  • Exponent constraint D^{272} L^{96} ≪ q^{11/16-ε} = Theorem 1.1 assumes D^{272} ≪ q^{11/16-1/2000}
    Ad hoc range guaranteeing the twisted first moment error terms are small; exact exponents inherited from [4].
  • Smoothing parameter Δ = Δ = sqrt(log log q)/log log log q
    Chosen in Section 6 to balance the approximation of log|L| by P(χχ_j) and the Beurling–Selberg error.
assumptions (6)
  • domain assumption Generalized Riemann Hypothesis (GRH)
    Explicitly assumed in Theorem 1.1, Lemma 4.4, Lemma 6.2; used to approximate L(1+2s, χ) by short Dirichlet polynomials.
  • ad hoc to paper Twisted first moment asymptotic (Theorem 3.1) from [4]
    Quoted from the authors' companion preprint [4, Theorem 1.3]; it is the load-bearing input for the random-model reduction (3.3).
  • ad hoc to paper High mollified moment bound (1.3) and φ*_F(q) ≍ q from [4]
    Used throughout (Lemmas 3.3, 6.2, Corollary 1.2) to control exceptional sets and normalize the weighted measure.
  • ad hoc to paper Cross-correlation bound for distinct characters (7.28) in [4]
    Used in Lemma 4.10 and Lemma 4.11 to show the off-diagonal terms contribute O(log log log q).
  • standard math Beurling–Selberg majorant (Lemma 6.3)
    Standard Fourier analysis tool used to approximate indicator functions of intervals.
  • standard math Orthogonality of Dirichlet characters modulo q
    Underlies the random-model identity (3.2) and the moment computations.

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Pith. "Pith review of Simultaneous non-vanishing of Dirichlet $L$--functions, II: Weighted central limit theorem." pith.science (2026). https://pith.science/paper/UPYVQRF4

@misc{pith2026260721532,
  author       = {Pith},
  title        = {Pith review of: Simultaneous non-vanishing of Dirichlet $L$--functions, II: Weighted central limit theorem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UPYVQRF4}},
  note         = {Machine review of arXiv:2607.21532}
}
abstract

Under the Generalized Riemann Hypothesis, we prove a weighted central limit theorem for the joint distribution of four Dirichlet $L$--functions at the central point, twisted by the family of primitive characters to a large prime modulus. As an application, we show that a positive proportion of the characters in the family yield four central values that are simultaneously large, and a positive proportion yield values that are simultaneously nonzero and small.

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Reference graph

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