Extreme values of quadratic Dirichlet $L$-functions

Hao Zhang; Shengbo Zhao; Weijia Wang; Zikang Dong

REVIEW 2 major objections 5 minor 1 cited by

Extreme values of quadratic Dirichlet $L$-functions

T0 review · 2 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read Assuming GRH, quadratic Dirichlet L-functions are shown to reach exp((1+o(1))√(log X log_3 X / log_2 X)), doubling the previous best constant from 1/2 to 1.

desk verdict The claimed improvement from 1/2 to 1 in the GRH-conditional extreme value bound for quadratic Dirichlet L-functions is undone by a sign error in the Rankin-trick step; as written, Theorem 1.1 does not follow, though the approach looks repairable. read the letter →

arxiv 2607.20408 v1 pith:IYGPTXWC submitted 2026-07-22 math.NT

Zikang Dong , Weijia Wang , Hao Zhang , Shengbo Zhao This is my paper · ORCID

classification math.NT MSC 11L4011M06

keywords quadraticDirichletL-functionsextremevaluesresonancemethodGCDsumsGeneralizedRiemannHypothesiscentralRankintricklowerbounds

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

The paper studies how large quadratic Dirichlet L-functions can be at their central point as the discriminant d grows. Under the Generalized Riemann Hypothesis, it proves that for sufficiently large X some fundamental discriminant d in [X,2X] satisfies |L(1/2, χ_d)| ≥ exp((1+o(1))√(log X log_3 X / log_2 X)). This improves the previous conditional constant from 1/2 to 1, matching the leading constant known for the Riemann zeta function. The method combines the resonance construction with large GCD sums and relies on a GRH-based mean-value estimate for quadratic characters.

What carries the argument

The central object is the GCD sum Σ_{m,n∈M} √((m,n)/[m,n]) over a set M of squarefree integers, paired with the resonator R_d = Σ_{n∈M} χ_d(n). Lemma 2.3 asserts the maximal GCD sum for |M|=N is N exp((2+o(1))√(log N log_3 N/log_2 N)). The proof also uses the approximate functional equation (Lemma 2.1) and a GRH-conditional mean-value estimate (Lemma 2.2) to compute the first and second moments of L(1/2,χ_d) R_d^2. The mechanism is that the second moment inherits the GCD sum of M while the first moment is essentially |M|, so their ratio produces the extreme value.

What would settle it

Check whether the inequality Σ_{r>X^ε} r^{1/2} ≤ X^{ε/6} Σ_r r^{1/3} holds; it does not for any r>X^ε, so the tail term in (3.7) is uncontrolled unless a different argument is supplied.

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Extended reading notes

Core claim

The paper's central claim is Theorem 1.1: under GRH, for large X, max_{X<|d|≤2X, d∈F} |L(1/2, χ_d)| ≥ exp((1+o(1))√(log X log_3 X / log_2 X)). The proof constructs a set M of squarefree integers with near-maximal GCD sums and defines a resonator R_d = Σ_{n∈M} χ_d(n). Expanding the weighted first and second moments, S_1 and S_2, and applying a GRH-conditional mean-value theorem (Lemma 2.2), the ratio S_2/S_1 is shown to be at least the GCD sum of M. With the optimal GCD sum from Lemma 2.3, this yields the stated lower bound.

Load-bearing premise

The proof relies on a step asserting that pairs with very large least common multiple contribute negligibly to the GCD sum; the inequality used for that step runs the wrong way, so the asserted bound is not actually derived.

Editorial extensions

If this is right

The extreme-value exponent for quadratic Dirichlet L-functions now matches the best known for the Riemann zeta function, indicating a common threshold across L-function families.
Under GRH, this is the sharpest lower bound known for |L(1/2,χ_d)| in terms of the leading constant.
The resonance plus GCD sum argument extends naturally to other families whose character sums admit similar mean-value theorems.
The proof gives a template for converting GCD-sum maxima into central-value lower bounds for real primitive characters.

Reading between the lines

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

If the flagged tail-bound inequality is repaired, the method could plausibly push the constant beyond 1, since the GCD sum construction is already at the conjectured optimum; the next bottleneck would be the moment error terms.
The dependence on GRH enters only through the mean-value estimate; an unconditional substitute for that estimate would immediately yield an unconditional (but weaker) extreme-value lower bound.
The technique might extend to higher moments or derivatives of L-functions, where the same GCD sum machinery could yield new lower bounds for central values and beyond.
A natural stress test is whether the same exponent appears for non-quadratic characters or for L-functions of higher rank, where the resonance form changes but the GCD sum may still dominate.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

This paper claims the expected improvement from 1/2 to 1 in the GRH-conditional extreme value problem for quadratic Dirichlet L-functions. The announced bound is new relative to Darbar–Maiti, and the architecture is the standard resonator-plus-GCD-sum method, with a sensible choice of N = X^{1/4−α}. The main term and error estimates follow established lines, and there is no circularity or fitting of constants to the result. If the proof were correct, this would be a clean within-field advance.

The trouble is in the Rankin-trick step leading to (3.7). The paper wants to lower-bound the restricted GCD sum over r = [m,n]/(m,n) ≤ X^ε by subtracting the tail from the full sum. It asserts the tail is at most X^{ε/6} Σ r^{-1/3}, but for r > X^ε we have r^{-1/2} = r^{-1/3} r^{-1/6} < X^{-ε/6} r^{-1/3}, so the tail is actually bounded by X^{-ε/6} Σ r^{-1/3}. The exponent sign is backwards. With the printed X^{ε/6}, the subtracted term is X^{ε/6} |M| exp(y_M^{2/3}), which for fixed ε>0 is exponentially larger than the main term N exp(2 sqrt(logN log3N / log2N)). Thus (3.7) cannot follow. This is the load-bearing step: Theorem 1.1 is an immediate corollary of (3.7), and without it the result is unproved. The correct sign would make the step plausible, and the rest of the proof would proceed.

The y_M bound from Lemma 2.3 is cited rather than proved, but that is secondary. The more serious issue is the wrong exponent, which is either a transcription mistake or a genuine gap. Either way, the manuscript in its present form should not be accepted.

I would still send this to a competent referee rather than desk reject, because the error is local and repairable, and the claimed improvement is the natural next step in the literature. If the authors fix the sign, the paper could be solid. As it stands, the central claim is not established.

Referee Report

2 major / 5 minor

Summary. Assuming GRH, the paper claims that for sufficiently large X, max_{X<|d|≤2X, d∈F} |L(1/2,χ_d)| ≥ exp((1+o(1)) sqrt(log X log_3 X / log_2 X)). This would improve the previous conditional constant 1/2 of Darbar and Maiti [5] to 1. The proof uses the resonance method: choose a set M of squarefree integers with large GCD sum (Lemma 2.3), set R_d = Σ_{n∈M} χ_d(n), and estimate S1 = Σ R_d^2 and S2 = Σ L(1/2,χ_d) R_d^2 via the approximate functional equation (Lemma 2.1) and a GRH mean-value lemma (Lemma 2.2). The ratio S2/S1 gives the lower bound. The key new ingredient is the Rankin-trick estimate (3.7) for the truncated GCD sum.

Significance. If correct, the theorem is a significant advance: it raises the conditional lower-bound constant for quadratic Dirichlet L-functions from 1/2 to 1, matching the best-known constant for ζ(s) in the Bondarenko–Seip line (though not the √2 of de la Bretèche–Tenenbaum). The method is a direct combination of existing tools, and the proof is short. The main novelty is the application of the large GCD-sum lemma together with the y_M bound. However, the proof as written contains a sign error in a key inequality; after correction, the argument is plausible but needs revision.

major comments (2)

[§3, derivation of (3.7)] The Rankin-trick step is written with the wrong exponent. For r=[m,n]/(m,n)>X^ε, r^{-1/2}=r^{-1/3}r^{-1/6} ≤ X^{-ε/6}r^{-1/3}, not ≥ X^{ε/6}r^{-1/3}. Therefore the tail should be bounded by X^{-ε/6}Σ r^{-1/3}, not X^{ε/6}Σ r^{-1/3}. As printed, X^{ε/6}Σ r^{-1/3} is of size N exp((2ε/3+o(1))log N), while the main term from Lemma 2.3 is only N exp(2√(log N log_3 N / log_2 N)); hence the right-hand side of (3.7) is not a lower bound. This step is load-bearing for Theorem 1.1. With the corrected exponent X^{-ε/6}, the subtracted term is negligible relative to the main term, so the claim appears repairable, but the displayed argument must be fixed and the subsequent estimates rechecked.
[Lemma 2.3 and §3] The proof of Theorem 1.1 uses not only Lemma 2.3 but also the fact that the extremal set M has y_M = max_{m∈M} P^+(m) ≤ (log N)^{1+o(1)}. This property is stated only in an informal note after Lemma 2.3, not as part of the lemma. It is used in the lower bound (3.6) and in the estimate Σ_{n∈M} ((m,n)/[m,n])^{1/3} ≤ exp(y_M^{2/3}). Please state the y_M bound explicitly in Lemma 2.3, with a proof or a precise reference, because without it the lower bound for D in (3.8) is incomplete.

minor comments (5)

[§3, S1 error term] The displayed error O(X^{1/2+ε}Σ_{m,n∈M}1) after Lemma 2.2 drops the factors g1(n1), g2(n2); these are not O(1) but exp((log n)^{1-ε}) and exp((log n)^{1/2-ε}). The argument should absorb them into X^ε or display the precise bound.
[After Lemma 2.2] There are notational inconsistencies: g2(n1) versus g2(n2), and the phrase 'It is clear for g1(n0)' should refer to n1 consistently.
[End of §3] The final 'Taking α→0+' should be formulated as a two-limit argument: for every η>0 choose α small enough, then X large enough. As written it may suggest a single limiting process in which the o(1) also varies.
[Abstract] The phrase 'as d is large' is awkward; use 'as |d|→∞'.
[Lemma 2.3 note] The informal note after Lemma 2.3 would be better integrated into the lemma statement so that the y_M property is a formal hypothesis for later use.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the claimed extreme value is a genuine output, not an input; imported lemmas are external.

full rationale

The proof derives a lower bound for max |L(1/2, chi_d)| by the resonance method. The target quantity is the output of the argument, never assumed as an input. The paper's main ingredients are external results: Lemma 2.2 is quoted from Darbar–Maiti [5, Lemma 2], Lemma 2.3 is the GCD-sum bound of de la Bretèche–Tenenbaum [4, Eq. (1.5)], and Lemma 2.1 is from Soundararajan [9]. None of these citations is authored by the present authors, so there is no self-citation chain. The resonator set M is taken from the external GCD-sum construction, and its properties (including y_M bound) are cited, not fitted to the paper's own conclusion. No parameter is fitted to data and then renamed a prediction. The possible sign error in the Rankin-trick inequality (3.7) is a correctness/mathematical-error concern, not a circularity concern: even if the inequality is wrong, the argument does not reduce to its own conclusion. Therefore the paper exhibits no definable circular step.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central claim is built on three imported results (approximate functional equation, GRH mean-value lemma, GCD-sum lower bound) and on asymptotic parameter choices α, ε. No new entities or fitted constants are introduced. The main unstated load is the moment/tail behaviour of the GCD-sum set M.

free parameters (2)

α = α→0+
Introduced in N=X^{1/4−α} to control error terms; tends to 0 in the final passage; not fitted to data.
ε = arbitrary small positive
Appears in Lemma 2.2 and in the restriction [m,n]/(m,n)≤X^ε; chosen per standard argument, not fitted.

assumptions (4)

domain assumption Generalized Riemann Hypothesis
Assumed in Theorem 1.1 and used in Lemma 2.2 for quadratic character sums.
domain assumption Lemma 2.2 (Darbar–Maiti mean-value estimate)
Imported from [5, Lemma 2]; supplies the GRH-conditional character-sum main term and error.
domain assumption Lemma 2.3 (La Bretèche–Tenenbaum GCD-sum lower bound)
Imported from [4, Eq. (1.5)]; provides the N exp(2√(...)) lower bound for the extremal GCD sum and the asserted y_M bound.
standard math Lemma 2.1 approximate functional equation
Imported from [9]; used to open the L-function into a smoothed Dirichlet series.

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Cite this review

Pith. "Pith review of Extreme values of quadratic Dirichlet $L$-functions." pith.science (2026). https://pith.science/paper/IYGPTXWC

@misc{pith2026260720408,
  author       = {Pith},
  title        = {Pith review of: Extreme values of quadratic Dirichlet $L$-functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IYGPTXWC}},
  note         = {Machine review of arXiv:2607.20408}
}

abstract

In this paper, we investigate extreme values of quadratic Dirichlet $L$-functions at the central point. We provide new extreme values of $L(\frac12,\chi_d)$ as $d$ is large, which improves the recent result of Darbar and Maiti.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Extreme values of quadratic Dirichlet $L$-functions with prime-related moduli
math.NT 2026-07 conditional novelty 4.0 of 10

Under GRH, max_{X<q≤2X} |L(1/2, χ_{8q})| ≥ exp((1+o(1)) sqrt(log X log_3 X / log_2 X)).

Reference graph

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Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [5]

On the frequency of Titchmarsh’s phenomenon for (s)-III , Proc

Balasubramanian, R.; Ramachandra, K. On the frequency of Titchmarsh’s phenomenon for (s)-III , Proc. Indian Acad. Sci. Sect. A , 86 (1977), 341--351

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[2] [1]

Lower bounds for the maximum of the Riemann zeta function along vertical lines , Math

Aistleitner, C. Lower bounds for the maximum of the Riemann zeta function along vertical lines , Math. Ann. , 365 (2016), 73--96

2016

[3] [2]

Large values of L -functions from the Selberg class , Journal of Mathematical Analysis and Applications , 446 (2017), 345--364

Aistleitner, C.; Pa\' n kowski, . Large values of L -functions from the Selberg class , Journal of Mathematical Analysis and Applications , 446 (2017), 345--364

2017

[4] [3]

Extreme values of the Riemann zeta function on the 1 -line , Int

Aistleitner, C.; Mahatab, K.; Munsch, M. Extreme values of the Riemann zeta function on the 1 -line , Int. Math. Res. Not. , 22 (2019), 6924--6932

2019

[5] [4]

On large values of L( , ) , Q

Aistleitner, C.; Mahatab, K.; Munsch, M.; Peyrot, A. On large values of L( , ) , Q. J. Math. , 70 (2019), 831--848

2019

[6] [6]

Large greatest common divisor sums and extreme values of the Riemann zeta function , Duke Math

Bondarenko, A.; Seip, K. Large greatest common divisor sums and extreme values of the Riemann zeta function , Duke Math. J. , 166 (2017), 685--701

2017

[7] [7]

Extreme values of the Riemann zeta function and its argument , Math

Bondarenko, A.; Seip, K. Extreme values of the Riemann zeta function and its argument , Math. Ann. , 372 (2018), 999--1015

2018

[8] [8]

Sommes de G\' a l et applications

de la Bret\` e che, R.; Tenenbaum, G. Sommes de G\' a l et applications. (French) [G\' a l-type sums and applications] , Proc. Lond. Math. Soc. , 119 (2019), 104--134

2019