REVIEW 3 major objections 3 minor 14 references
Extreme values of quadratic Dirichlet $L$-functions with prime-related moduli
T0 review · 3 major / 3 minor · reviewed 2026-08-01 · deepseek-v4-flash
Pith's one-line read Assuming GRH, some prime q in (X,2X] makes |L(1/2, χ_{8q})| as large as exp((1+o(1))√(log X log log log X / log log X)).
desk verdict The main theorem is likely true and the constant 1 is a genuine improvement over Gao's 1/2, but the written proof needs two small repairs before it is referee-ready. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The machinery is the resonance method with a GCD-sum amplifier. The central object is the resonator R_q = Σ_{m∈M} χ_{8q}(m), where M is a set of square-free integers of size X^{1/4-δ} maximizing the GCD sum Σ_{m,n∈M} √((m,n)/[m,n]). Two estimates carry the proof: Lemma 2.2, a GRH-conditional bound ∑_{q}^{*}(log q)χ_{8q}(c)Φ̂(q/X) = 1_{c=□}Φ̂(1)X + O(X^{1/2+ε} log(c+2)) for odd c, which isolates the diagonal main terms; and Lemma 2.3, a lower bound for GCD sums of square-free integers, which provides the exponential growth. The restriction [m,n]/(m,n) ≤ X^ε, enforced by Rankin's trick, lets the smooth weight ω be treated as 1.
What would settle it
Evaluate the character sum in Lemma 2.2 for c = 4: since χ_{8q}(4) = 0 for every odd prime q, the sum is 0, whereas the lemma's asserted main term would be Φ̂(1)X (because 4 is a square). This shows the lemma cannot be applied to even c, and thus the resonator set M in the proof must be restricted to odd integers for the diagonal main term S1 ≈ Φ̂(1)XN to be correct.
Extended reading notes
Core claim
Theorem 1.1 states that, assuming GRH, for sufficiently large X, max_{X<|q|≤2X} |L(1/2, χ_{8q})| ≥ exp((1+o(1))√(log X log_3 X / log_2 X)). The proof constructs a resonator R_q = Σ_{m∈M} χ_{8q}(m), with M a set of N = X^{1/4-δ} square-free integers chosen to maximize the GCD sum Σ_{m,n∈M} √((m,n)/[m,n]). The core of the argument is the ratio S2/S1: S1 counts the diagonal terms m=n via a conditional character-sum estimate (Lemma 2.2), and S2 is bounded below using the same estimate plus the observation that the smooth weight ω is close to 1 when [m,n]/(m,n) ≤ X^ε. The GCD-sum bound of Lemma 2.3 gives the exponential factor, and choosing δ → 0 yields the constant 1.
Load-bearing premise
The proof's load-bearing premise is that a GRH-conditional estimate for sums of χ_{8q}(c) over primes (Lemma 2.2) holds uniformly for every odd c up to exp((log N)^{1+o(1)}); the paper uses it for c = mn and c = kmn without expressly requiring M to be odd, and for even c the estimate's main term is false.
Editorial extensions
If this is right
- Under GRH, the family {χ_{8q}} of prime-related moduli has extreme values of the same asymptotic size as the full family of quadratic Dirichlet L-functions, with constant 1 instead of 1/2.
- The proven lower bound grows faster than any fixed power of log X, so |L(1/2, χ_{8q})| can be exceptionally large even when the conductor is restricted to 8 times a prime.
- The resonance method remains effective on a thin subfamily of conductors, provided the conditional character-sum estimate holds uniformly for all odd c up to exp((log N)^{1+o(1)}).
- The GCD-sum bound is the sole source of the exponential factor; improvements to that bound would directly improve the constant in the lower bound.
Reading between the lines
- The proof applies Lemma 2.2 to c = mn and c = kmn, but the lemma is stated only for odd c; since χ_{8q}(even) = 0, applying it to even c would give a wrong main term. The set M must therefore be chosen to contain only odd integers, a condition the paper does not explicitly state — a repairable but real gap in the written proof.
- The same construction could be tested for moduli of the form d·q with fixed squarefree d; the character-sum estimate would need adjustment, but the GCD-sum part should transfer unchanged.
- Because the error bounds require log(c+2) ≪ X^ε, the method implicitly constrains M to elements of size exp((log N)^{1+o(1)}); a resonator with larger elements would break the uniformity of Lemma 2.2.
- A direct computation of the character sum in Lemma 2.2 at c = 4 would confirm that the main term is zero, not Φ̂(1)X, illustrating why the oddness of M is necessary.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proves, under GRH, a lower bound of the form max_{X<|q|≤2X} |L(1/2, χ_{8q})| ≥ exp((1+o(1)) sqrt(log X log_3 X / log_2 X)), improving the constant 1/2 obtained by Gao for the same family of quadratic Dirichlet L-functions with prime-related moduli. The method is the resonance method: a resonator R_q = Σ_{m∈M} χ_{8q}(m) is constructed from a set M of squarefree integers with large GCD sum (Lemma 2.3), and the ratio S_2/S_1 is bounded below using a GRH-conditional character-sum estimate (Lemma 2.2) and the approximate functional equation (Lemma 2.1). The proof follows the standard pattern of Bondarenko-Seip and de la Bretèche-Tenenbaum adapted to this family.
Significance. If the result is correct, it constitutes a genuine improvement over Gao's recent constant 1/2 and matches the constant 1 that is expected from the resonance/GCD-sum method for this family. The argument is not self-contained: it relies on two substantial external inputs, Lemma 2.2 from Gao's preprint and Lemma 2.3 from de la Bretèche-Tenenbaum, but such reliance is normal in this area. The paper's main contribution is the correct adaptation of the GCD-sum method to the moduli 8q, and the result is likely to be of interest to specialists. However, as written, the proof contains two load-bearing technical gaps that prevent the main estimate from being fully justified.
major comments (3)
- [§3, before (3.3) and after (3.4)] Lemma 2.2 is stated only for positive odd integers c. In the evaluation of S_1, Lemma 2.2 is applied with c=mn, and in S_2 with c=kmn. The set M is taken as in Lemma 2.3, which only requires squarefree integers and does not guarantee that m,n are odd. If an even m∈M exists, then the diagonal term m=n gives c=m^2 even, for which the displayed main term in Lemma 2.2 is false (indeed χ_{8q}(m^2)=0 for odd primes q). Similarly, in S_2 the index k must be restricted to odd integers. This is not a mere formality: the main terms in (3.3) and (3.4) would be incorrect for even c, and the final ratio would change. The likely repair is to choose M to consist of odd squarefree integers, as can be done in the construction of [5], and to sum over odd k in S_2. The authors should state this explicitly and verify the consequences for M's size and GCD-sum bound.
- [§3, Eq. (3.3)] The equation S_1 = Φ̂(1)XN + O(X^{1/2+2ε}N^2) ≤ (1+o(1))Φ̂(1)X is inconsistent: since N=⌊X^{1/4-δ}⌋→∞, the displayed upper bound omits the factor N and is false. The correct bound is S_1 ≤ (1+o(1))Φ̂(1)XN. This error is consequential because if S_1 were only O(X), the ratio S_2/S_1 would gain an extra factor N, artificially changing the final exponent. In the subsequent text the correct normalization appears to be used implicitly, but as printed the proof is self-contradictory. This must be corrected before the claimed lower bound can be accepted.
- [§3, passage after (3.4)] The proof of the lower bound for S_2 depends crucially on Lemma 2.2 being valid uniformly for c=kmn with k up to X^{1/2+ε} and m,n∈M, whose elements may be as large as exp((log N)^{1+o(1)}). The paper imports Lemma 2.2 from [7] without proof and without checking uniformity conditions. In particular, the statement of Lemma 2.2 has an error term X^{1/2+ε} log(c+2), and the number of such c is large. The authors should either quote the precise uniformity range from [7] or prove the needed estimate. This is not a circularity (the external lemma is not fitted to the conclusion), but it is a load-bearing dependency that the manuscript should make explicit and verify.
minor comments (3)
- [§3, after (3.4)] "By Lemma (2.2), ω(ξ) and ω'(ξ) decay exponentially" should refer to Lemma 2.1, not Lemma 2.2.
- [Notation throughout] The introduction says q always represents an odd prime, but later uses both X<q≤2X and X<|q|≤2X. Since q is positive, |q| is redundant; unify the notation.
- [References] The author list in [10] appears garbled: "H. Z. Z. Dong, W. Wang and S. Zhao" should likely list the four authors consistently with the title page. Please correct.
Circularity Check
No circular derivation: the proof is a resonance-method application of external GRH character-sum and GCD-sum lemmas; the sole self-citation is background only.
full rationale
The argument is a resonance-method lower bound: Theorem 1.1 follows from the ratio S2/S1 in (3.1). The estimates of S1 and S2 are reductions to two external results: Lemma 2.2, a GRH-conditional character-sum estimate explicitly quoted from Gao's preprint [7] ('Proof. This is [7, Lemma 2.4].'), and Lemma 2.3, the Gál/GCD-sum theorem of de la Bretèche and Tenenbaum ('Proof. This is [5, Eq. (1.5)]'). Neither is derived from the conclusion of Theorem 1.1, and neither is fitted to the quantity being bounded. The resonance polynomial R_q and the smooth weight Φ are standard constructions, not data-derived fits. The self-citation [10] appears only in the Introduction ('In earlier work, the authors [10] improved the constant 1/2 to 1.') and is never used in the proof; it is not load-bearing. The manuscript does not invoke a uniqueness theorem, does not rename an empirical pattern, and does not define its object in terms of the result. Concerns about whether Lemma 2.2's odd-c hypothesis is satisfied when applied to c=mn and c=kmn, and whether (3.3) should read XN, are gaps in the written proof; they do not make the claimed lower bound equal to an input by construction. Hence no circular step can be exhibited; the only mild concern is the non-load-bearing self-citation, which is why the score is 2 rather than 0.
Assumptions & free parameters
assumptions (4)
- domain assumption Generalized Riemann Hypothesis (GRH)
- domain assumption Conditional character-sum estimate (Lemma 2.2, from [7])
- standard math GCD-sum maximum estimate (Lemma 2.3, from [5, Eq. (1.5)])
- domain assumption M is odd
Cite this review
Pith. "Pith review of Extreme values of quadratic Dirichlet $L$-functions with prime-related moduli." pith.science (2026). https://pith.science/paper/QNNQRSGU
@misc{pith2026260725403,
author = {Pith},
title = {Pith review of: Extreme values of quadratic Dirichlet $L$-functions with prime-related moduli},
year = {2026},
howpublished = {\url{https://pith.science/paper/QNNQRSGU}},
note = {Machine review of arXiv:2607.25403}
}
abstract
In this paper, we show a new lower bound for extreme values of quadratic Dirichlet $L$-functions with prime-related moduli, which generalizes the work of Darbar and Maiti in 2025, and sharpens a recent work of Gao in 2026.
Reference graph
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Reviewed August 1, 2026 · model on record in the stance chip above.
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