Large Value Estimates for Dirichlet Polynomials with Characters and Zero Density of Dirichlet $L$-Functions

Bin Chen; Vishal Gupta; Yung Chi Li

REVIEW 2 major objections 6 minor 21 references

Large Value Estimates for Dirichlet Polynomials with Characters and Zero Density of Dirichlet $L$-Functions

T0 review · 2 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper proves the zero-density bound (qT)^{7(1-σ)/3+ε} for Dirichlet L-functions summed over characters modulo q, improving the earlier exponent 12/5 and yielding new bounds on least primes and Goldbach numbers.

desk verdict A serious Guth-Maynard extension to Dirichlet characters with a plausible 7/3 exponent, but a systematic N^{σ/6} vs N^σ threshold typo must be fixed before the estimates as written are trustworthy. read the letter →

arxiv 2507.08296 v2 pith:6F5SNWWQ submitted 2025-07-11 math.NT

Bin Chen , Vishal Gupta , Yung Chi Li This is my paper · ORCID

classification math.NT MSC 11M0611M2611N0511N13

keywords largevalueestimatesGuth-Maynardmethodzero-densityDirichletL-functionspolynomialsdistributionofprimesinarithmeticprogressionsGoldbachnumber

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 proves a new zero-density estimate for Dirichlet $L$-functions: summed over all characters $\chi$ modulo $q$, the number of zeros $\rho=\beta+it$ of $L(s,\chi)$ with $\beta\ge\sigma$ and $|t|\le T$ is bounded by $(qT)^{7(1-\sigma)/3+\epsilon}$, improving the earlier exponent $12/5$. The proof extends a recent large-value method for long Dirichlet polynomials to polynomials twisted by primitive characters, and its key new ingredient is a sharp bound for sums over affine transformations of smooth functions carrying a GCD twist. From the new density bound the paper derives two arithmetic consequences: a bound on the least prime in an arithmetic progression modulo a prime power, and a bound on the least Goldbach number in a progression modulo a prime. Zero-density exponents control how many zeros can sit close to $\sigma=1$, and tighter control translates directly into sharper quantitative statements about primes.

What carries the argument

The load-bearing object is the large-value set $W$ of separated pairs $(t,\chi)$ on which a character-twisted Dirichlet polynomial $D_N(t,\chi)=\sum_{N<n\le 2N}a_n\chi(n)n^{it}$ is large. The argument forms the $|W|\times N$ matrix $M$ with entries $w(n/N)\chi(n)n^{it}$, bounds its largest singular value through the traces of $MM^*$ and $(MM^*)^3$, and splits the resulting oscillatory sum into $S_1,S_2,S_3$ according to how many of the three Fourier variables vanish. $S_3$, the hardest term, is controlled by the paper's principal new tool, Proposition 8.1: a bound for the integral over $u$ of the square of $\sum_{b,m_1,m_2,m_3}f_b((m_1u+m_3)/m_2)\,(am_1+bm_2+m_3,q)$, valid when the smooth functions $f_b$ have Fourier decay $|\hat f_b(\xi)|\lesssim (qT)^3(T/|\xi|)^j$. This affine-transform-with-GCD-twist estimate is what carries the adaptation of the Guth-Maynard large-value method (a recent technique for bounding how often a long Dirichlet polynomial can be large) into the character setting.

What would settle it

Take an explicit small modulus $q$, a separated set $W$ of pairs $(t,\chi)$, and the functions $f_b(u)=\psi(u)|\widetilde R_{M_2}(u,-b)|^2$ of Section 11, then compute $|\hat f_b(\xi)|$ at frequencies $|\xi|$ from $T$ up to $(qT)^2$: if the decay is ever weaker than $(qT)^3(T/|\xi|)^j$, the hypothesis of Proposition 8.1 fails and the $S_3$ bound — hence the exponent $7/3$ — collapses.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that a recent large-value method for long Dirichlet polynomials survives the passage to polynomials twisted by primitive Dirichlet characters, provided the character sums are controlled by a new bound for affine transformations with GCD twists. For a separated set $W$ of pairs $(t,\chi)$ on which a character-twisted polynomial of length $N$ attains values $\ge V$, Theorem 1.3 gives $|W|\ll_\epsilon N^2V^{-2}+(qT)^{4/3}N^2V^{-4}$ when $(qT)^{3/4}\le N\le (qT)^{5/6}$, and a further four-term bound for larger $N$. Fed into the zero-detection method, this yields $\sum_{\chi\bmod q}N(\sigma,T,\chi)\ll_\epsilon (qT)^{4(1-\sigma)/(1+\sigma)}$, and the combination with the classical bound for $\sigma\le 5/7$ produces the headline estimate $(qT)^{7(1-\sigma)/3+\epsilon}$. The paper also shows the new exponent pays off arithmetically: for fixed prime $p$, the least prime $p(p^n,k)$ in the progression $k\bmod p^n$ is $\ll_{p,\epsilon}(p^n)^{7/3+\epsilon}$, and the least Goldbach number $G(p,k)\equiv k\pmod p$ is $\ll_\epsilon p^{7/6+\epsilon}$.

Load-bearing premise

The $S_3$ bound, and with it the entire improved exponent $7/3$, rests on the Fourier decay assumption that the smoothed functions $f_b(u)=\psi(u)|\widetilde R_{M_2}(u,-b)|^2$ satisfy $|\hat f_b(\xi)|\lesssim (qT)^3(T/|\xi|)^j$ for every $j$; Section 11 asserts this decay in a single sentence via the convolution theorem, and the derivative estimates behind it are not written out.

Editorial extensions

If this is right

Corollary 1.5: for a fixed prime $p$, every integer $k$ coprime to $p^n$ has a prime in the progression with $p(p^n,k)\ll_{p,\epsilon}(p^n)^{7/3+\epsilon}$.
Corollary 1.6: for an odd prime $p$, the least Goldbach number $G(p,k)\equiv k\pmod p$ satisfies $G(p,k)\ll_\epsilon p^{7/6+\epsilon}$.
At the threshold $N=(qT)^{4/5}$ with $V=N^{3/4}$, Theorem 1.3 gives $|W|\ll(qT)^{8/15}$, strictly better than the $(qT)^{3/5}$ from both the classical mean value theorem and the Halász–Montgomery–Huxley estimates.
In the range $5/7\le\sigma\le 7/9$ the new bound is the strongest available zero-density estimate for all $q,T$ simultaneously, since it beats (1.3) for $\sigma>5/7$ and (1.4) for $\sigma<7/9$.
The abstract additionally announces new results on primes in arithmetic progressions in short intervals, in particular for prime-power moduli, as consequences of the density estimate.

Reading between the lines

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

Beyond the paper: the Fourier decay asserted in Section 11 is the point where the proof is most exposed; writing out the full derivative estimates — or exhibiting a configuration where they fail — would clarify whether the method still has slack to push below the exponent $7/3$.
Beyond the paper: the GCD-twist affine-sum bound of Proposition 8.1 is also the natural lever for the $Q^2T$-analogue that the introduction points to; averaging over moduli up to $Q$ rather than a single $q$ would likely need a $Q$-dependent version of this proposition.
Beyond the paper: the same architecture of singular values, trace expansion, and affine-transform sums should transfer to other $L$-function families whose characters enter through sums of the same shape, so the improvement need not stop at Dirichlet $L$-functions.
Editorial note: Corollary 1.6 attributes its antecedent to 'Jutila [7]', but entry [7] in the reference list is Davies's Kakeya paper; the Jutila work on the least Goldbach number is listed as [15]. This mismatch does not touch the proof of the corollary.

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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

Quick take: this paper extends the Guth–Maynard large value machinery to Dirichlet characters and claims a zero-density exponent of 7/3. The new content is real—the affine-transformation/GCD-twist sum in Proposition 8.1 is a genuine adaptation of the method—and the overall architecture follows the Guth–Maynard template cleanly. If the main estimates hold, it improves Huxley's 12/5, Jutila's, and Forti–Viola's exponents, and the applications to the least prime and Goldbach numbers are reasonable corollaries.

The problem is a threshold error that runs through the core lemmas. Lemma 4.1, Proposition 3.1, Proposition 4.6, and Proposition 10.1 all state |S_N(t,χ)| ≥ N^{σ/6}, but the SVD argument gives |W| ≲ N^{1−σ/3} s1(M)^2, not N^{1−2σ} s1(M)^2. The stress-test note is right: with the weaker threshold, the energy bound and the final large-value estimate collapse. The intended threshold is clearly N^σ—Section 12.3 uses |D_N| ⪆ N^σ—so this looks like a systematic typo rather than a conceptual gap. Still, it has to be fixed all through Sections 3, 4, 10, and 11, and the exponents re-checked at each step. A referee should demand that.

The other soft spot is the verification of the Fourier decay hypothesis for Proposition 8.1 in Section 11. They assert the decay in a single sentence using the convolution theorem and the |W|² bound. That step needs the derivative estimates written out; without them, S3 and the whole theorem depend on an unproven hypothesis. It may be routine, but it is not routine from what is printed.

The rest of the paper looks credible. The S1 and S2 bounds follow the established pattern with Heath-Brown's double zeta sum bound, and the energy argument is structurally sound. There are some citation slips (e.g., Corollary 1.6 cites [7] for Jutila, which is Davies), but those are cosmetic.

Bottom line: this is a serious paper with a plausible main theorem, but not ready as written. I would send it to a strong referee and ask for the threshold fix and the Fourier-decay details before recommending acceptance. It deserves the referee's time.

Referee Report

2 major / 6 minor

Summary. The paper adapts the Guth-Maynard large-value method to Dirichlet polynomials twisted by primitive Dirichlet characters and obtains a zero-density estimate for Dirichlet L-functions: summing N(σ,T,χ) over χ mod q gives exponent 7/3, improving Huxley's 12/5. The proof follows the standard framework: a matrix SVD reduction, bounds for the trace terms S1, S2, S3, an affine-GCD-twist sum estimate, moments and energy estimates for the character-weighted function R, and a final zero-detection argument. Two arithmetic applications are stated: an improved bound for the least prime in an arithmetic progression modulo a prime power and an improved bound for the least Goldbach number modulo a prime.

Significance. If correct, this is a substantial advance: it is the first adaptation of the Guth-Maynard machinery to character-twisted Dirichlet polynomials and yields the best known zero-density exponent in the q-aspect as well as in T. A notable strength is that the argument is parameter-free: the bounds are derived from analytic lemmas (Heath-Brown's double zeta sum bound, Guth-Maynard's matrix lemmas) rather than tuned to the final exponent. The claimed improvement from 12/5 to 7/3, together with the arithmetic corollaries, makes this a consequential contribution to multiplicative number theory, provided the normalization and Fourier-decay issues described below are resolved.

major comments (2)

[Sections 3, 4, 10; Lemma 10.3; Section 12.3] The threshold N^{σ/6} is internally inconsistent with the SVD argument and with the rest of the paper. In Lemma 4.1 the hypothesis |S_N(t,χ)| ≥ N^{σ/6} and the inequality ∑_{(t,χ)} |S_N(t,χ)|^2 ≤ s_1(M)^2 ∑_n |b_n|^2 give |W| N^{σ/3} ≤ N s_1(M)^2, hence |W| ≤ N^{1−σ/3} s_1(M)^2, not |W| ≤ N^{1−2σ} s_1(M)^2. For σ = 4/5 the claimed right-hand side N^{−3/5} s_1(M)^2 is typically below 1, which cannot bound a nonempty set W. Lemma 10.3's proof and Section 12.3 both use the condition |D_N(t,χ)| ≥ N^σ, and Proposition 10.1 is used with the same normalization in Section 11. The intended hypothesis is evidently V ≈ N^σ, up to an absolute constant (e.g. N^σ/6, not N^{σ/6}). The occurrences in Lemma 4.1, Propositions 3.1, 4.6, and 10.1, and in the S3/energy bounds of Sections 10–11, must be corrected systematically, and the reduction in Section 3 from |D_N| ≥ N^σ to the three parts D^{(i)} each inheriting ≥ N^σ/3 should be restated with the corrected '≥ c N^σ' form.
[Section 11.1, Proposition 8.1 hypothesis] The verification that f_b(u) = ψ(u)|\tilde R_{M2}(u,-b)|^2 satisfies the Fourier-decay hypothesis \hat f_b(ξ) ≤ (qT)^3 (T/|ξ|)^j is compressed into a single sentence. A complete proof needs to use the convolution theorem for \tilde R_{M2}^2 as K * |R|^2, then handle the product with ψ by splitting the convolution integral for \hat ψ * \widehat{|\tilde R|^2} at |η| ≈ |ξ|/2, and it should quote the correct bound on ∫|R|^2 from Lemma 9.1 (which is ≲ ϕ(q)|W|, hence ≤ (qT)^{O(1)}, rather than literally |W|^2). Since Proposition 8.1 is the key input to the S3 bound, the omitted derivation should be written out.

minor comments (6)

[Proposition 4.6] The statement writes 'W ≲_ϵ ...' where it should write '|W| ≲_ϵ ...'.
[Section 3, proof of Theorem 1.3] In the splitting argument, the third part is said to give the same bound for '|W2|' again; it should be '|W3|'.
[Sections 1 and 3] The reference to Guth-Maynard's work appears as '[8]' in Theorem 1.3, but in the bibliography [8] is Forti-Viola and [9] is Guth-Maynard; the citation should be [9].
[Section 12.3, first range] When applying the first part of Theorem 1.3 to D_N^2, the first term should be N^{4−4σ} rather than N^{2−2σ}; the subsequent domination by the second term is unaffected, but the displayed formula should be corrected.
[Throughout] There are numerous typos and formatting issues, including 'V ALUE' in the title, 'Specificallly', 'definded', 'triangel', 'Cauchy-Schwaz', 'innner sum', 'Airth', and 'Moebius'; these should be cleaned up before publication.
[Section 11.1] The sentence 'Applying the convolution theorem and noting ∫|R(u,−b)|^2 1_{[1/2,2]}(u) du ≲ |W|^2 ≤ (qT)^2' should be replaced by the precise Lemma 9.1 bound ≲ φ(q)|W|, with a short explanation of why this is sufficient for the Fourier-decay hypothesis.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation found: the main bounds are proved from independent analytic lemmas, not from fitted or self-referential inputs.

full rationale

The paper derives Theorem 1.4 by combining a new large-value estimate (Theorem 1.3) with the standard zero-detection method; Theorem 1.3 is itself proved from SVD/trace identities, Poisson summation, Heath-Brown's double zeta sum bound, and an extension of the Guth-Maynard S3 estimates. No step fits a parameter to the final zero-density bound and then renames it a prediction. The only cited author self-work in the references (Chen-Debruyne-Vindas, reference [5]) is not used as a load-bearing ingredient in the main proof. The brief verification in Section 11 that the functions f_b have the Fourier decay needed for Proposition 8.1 is compressed, but the claimed decay is not derived from the large-value or zero-density conclusions it is used to prove; it is a quantitative check requiring a fuller justification, not circularity. Similarly, the apparent threshold mismatch between N^(sigma/6) in Lemma 4.1 and N^sigma in Section 12.3 is an exponent-consistency issue, not an instance of assuming the target estimate. The derivation is therefore self-contained, and no circular step can be exhibited.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No fitted parameters and no invented entities. The derivation imports standard analytic facts and two structural assumptions: the transfer of Guth-Maynard lemmas to the character setting, and the Fourier decay hypothesis of Proposition 8.1. These are the main unproved inputs.

assumptions (3)

domain assumption Standard analytic number theory facts: approximate functional equation, fourth moment bound for L(s,chi), and zero-spacing estimate N(sigma,T+1,chi)-N(sigma,T,chi) of size at most log(qT).
Used in Section 12 for the zero-detection method and class-II zero bound; taken from Davenport and Montgomery as standard literature.
ad hoc to paper Fourier decay hypothesis of Proposition 8.1: hat f_b(xi) is at most (qT)^3 (T/|xi|)^j for the smoothed functions f_b = psi |tilde R_{M2}|^2.
This condition is the input to the key S3 estimate. The verification in Section 11 is stated in one sentence rather than fully demonstrated.
ad hoc to paper The Guth-Maynard lemmas on singular values, trace expansions, and energy bounds transfer to the character-twisted setting without loss of strength.
Sections 4 through 10 repeatedly say 'adapting [9, Lemma/Section]'; the transfer is plausible but not fully self-contained in this preprint.

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Cite this review

Pith. "Pith review of Large Value Estimates for Dirichlet Polynomials with Characters and Zero Density of Dirichlet $L$-Functions." pith.science (2026). https://pith.science/paper/6F5SNWWQ

@misc{pith2026250708296,
  author       = {Pith},
  title        = {Pith review of: Large Value Estimates for Dirichlet Polynomials with Characters and Zero Density of Dirichlet $L$-Functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6F5SNWWQ}},
  note         = {Machine review of arXiv:2507.08296}
}

abstract

It is proved that \[ \sum_{\chi \bmod q}N(\sigma,T,\chi) \ll_{\epsilon} (qT)^{7(1-\sigma)/3+\epsilon}, \] where $N(\sigma,T,\chi)$ denotes the number of zeros $\rho=\beta+it$ of $L(s,\chi)$ in the rectangle $\sigma\leq \beta\leq 1$, $|t|\leq T$. The exponent $7/3$ improves upon Huxley's earlier exponent of $12/5$. The key innovation lies in deriving a sharp upper bound for sums over affine transformations of functions with a GCD twist, which arises from our adaptation of the Guth--Maynard method. As applications of the zero density estimates obtained in this paper, we derive a new upper bound for the least Goldbach number in arithmetic progressions modulo a prime and establish new results on primes in arithmetic progressions in short intervals, in particular for prime-power moduli.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

21 extracted references · 19 canonical work pages

[7]

R. O. Davies, Some remarks on the Kakeya problem, Proc. Cambridge Philos. Soc, 69 (1971), 417–421

work page 1971
[1]

M. B. Barban, YU. Y. Linnik, N. G. Tshudakov, On prime numbers in an arithmetic progression with a prime-power difference, Acta Airth, 9 (1964), 375–390

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[5]

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arXiv

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [7]

R. O. Davies, Some remarks on the Kakeya problem, Proc. Cambridge Philos. Soc, 69 (1971), 417–421

work page 1971

[2] [1]

M. B. Barban, YU. Y. Linnik, N. G. Tshudakov, On prime numbers in an arithmetic progression with a prime-power difference, Acta Airth, 9 (1964), 375–390

work page 1964

[3] [2]

Bourgain, On large values estimates for Dirichlet polynomials and the density hypothesis for the Riemann Zeta function, Int

J. Bourgain, On large values estimates for Dirichlet polynomials and the density hypothesis for the Riemann Zeta function, Int. Math. Res Not. 3 (2000), 133–146

work page 2000

[4] [3]

Bourgain, On the distribution of Dirichlet sums

J. Bourgain, On the distribution of Dirichlet sums. II, in: Number theory for the millennium, I (Urbana, IL, 2000), pp. 87–109, A K Peters, Natick, MA, 2002

work page 2000

[5] [4]

Bourgain, Remarks on Montgomery’s conjectures on Dirichlet series, Geometric Aspects of Functional Analysis (1989–1990), Springer Lecture Notes in Mathematics 1469 (1991), 153–165

J. Bourgain, Remarks on Montgomery’s conjectures on Dirichlet series, Geometric Aspects of Functional Analysis (1989–1990), Springer Lecture Notes in Mathematics 1469 (1991), 153–165

work page 1991

[6] [5]

B. Chen, G. Debruyne, J. Vindas, On the density hypothesis for L-functions associated with holomorphic cusp forms, Rev. Mat. Iberoam. 40 (2024), 2179-2202

work page 2024

[7] [6]

Davenport, Multiplicative number theory, third ed., Springer-Verlag, New York, 2000

H. Davenport, Multiplicative number theory, third ed., Springer-Verlag, New York, 2000. 50 B. CHEN

work page 2000

[8] [8]

Forti, C

M. Forti, C. Viola, Density estimates for the zeros of L-functions, Acta Arith. 23 (1973), 379–391

work page 1973