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A Selberg-type zero-density result for twisted $\rm GL_2$ $L$-functions and its application

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read An unconditional zero-density estimate for twisted GL2 L-functions yields Gaussian statistics for their argument as the character varies.

desk verdict Plausible and novel application of Selberg's machinery to twisted GL2 L-functions, but the proof of the zero-density theorem has a serious unverified hypothesis in the argument-principle step. read the letter →

arxiv 2505.23573 v1 pith:SEU2P64J submitted 2025-05-29 math.NT

Qingfeng Sun , Hui Wang , Yanxue Yu This is my paper
classification math.NT MSC 11F1211F66
keywords zero-densityestimatestwistedL-functionsargumentfunctionmomentsofS(t)centrallimittheoremholomorphiccuspformsDirichletcharactersmollifiedsecondmoment
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

For a fixed holomorphic cusp form $f$, the paper proves an unconditional zero-density estimate of the classical argument-principle type for the family of twisted $L$-functions $L(s, f\otimes\chi)$ as $\chi$ runs over primitive Dirichlet characters modulo an odd prime $q$: on average over $\chi$, the number of zeros with real part at least $\sigma \ge 1/2 + 1/\log q$ in any vertical window of length at least $1/\log q$ is bounded by $(1+|t_1|+|t_2|)^A (t_2-t_1)\, q^{-\eta(\sigma-1/2)}\log q$, with an absolute constant $A$ and a positive exponent $\eta$. Such a bound, which requires no unproved hypothesis, is the kind of zero control that classically turns into statistical statements about arguments of $L$-functions. The paper then derives the even moments of $S(t, f\otimes\chi) = \pi^{-1}\arg L(1/2+it, f\otimes\chi)$ averaged over $\chi$ — they are $\frac{(2n)!}{n!(2\pi)^{2n}}(\log\log q)^n$ plus a smaller error — and a central limit theorem: $S(t, f\otimes\chi)/\sqrt{\log\log q}$ converges in distribution to a normal law with variance $(2\pi^2)^{-1}$. The conclusion is that, as the twisting character varies, the argument of a single twisted $\mathrm{GL}_2$ $L$-function at a fixed height is asymptotically Gaussian.

What carries the argument

The engine is an argument-principle wrapper applied to the holomorphic function $\omega(s) = 1 - (L(s,f\otimes\chi)M(s,f\otimes\chi) - 1)^2$, where $M$ is a mollifier of length $L = q^c$ with $0<c<1/360$ built from the convolution inverse of the Hecke eigenvalues. The zeros of $\omega$ are zeros of the $L$-function, yet its logarithmic integrals are controlled exactly by the averaged second moment $\frac{1}{\varphi^*(q)}\sum_{\chi}^* |LM-1|^2$, and Lemma 2.4 supplies the required bound $(1+|t|)^A q^{-\eta(\sigma-1/2)}$ uniformly for $\sigma \ge 1/2 - 1/\log q$. The exponential decay in $q$ is what forces zeros off the critical line to be rare on average. For the moments, the paper approximates $S(t,f\otimes\chi)$ by the imaginary part of a Dirichlet polynomial over primes $p \le x^3$ with $x = q^{\delta/3}$, and uses the zero-density estimate to control the error made by the $\sigma_x$-shifted line, where $\sigma_x$ is $1/2$ plus the maximal zero excursion near height $t$.

What would settle it

Compute numerically, for a fixed cusp form such as the discriminant function $\Delta$ and primes $q$ up to a few thousands, the averaged mollified second moment $\frac{1}{\varphi^*(q)}\sum_{\chi}^* |L(\sigma+it, f\otimes\chi)M(\sigma+it, f\otimes\chi) - 1|^2$ at $\sigma = 1/2 + (\log\log q)/\log q$, $t=0$, with mollifier length $L = \lfloor q^{1/400}\rfloor$. Lemma 2.4 forces this average to decay like $(\log q)^{-\eta}$; if it instead stays bounded away from zero or grows with $q$, the central estimate, and with it Theorems 1.1 and 1.2, would be false.

Watch

Extended reading notes

Core claim

The paper's central claim is Theorem 1.1: for a fixed primitive holomorphic cusp form $f$ of even weight and level $r$ with trivial nebentypus, and for an odd prime $q$ coprime to $r$, the average $N(f;\sigma,t_1,t_2)$ over primitive characters $\chi \bmod q$ of the number of zeros of $L(s,f\otimes\chi)$ with real part $\ge \sigma$ and imaginary part in $[t_1,t_2]$ satisfies $$N(f;\$\sigma$,t_1,t_2) \ll (1+|t_1|+|t_2|)^A (t_2-t_1)\, $q^{{-\eta(\sigma-1/2)}}$\log q$$ for $\sigma \ge 1/2 + 1/\log q$ and $t_2-t_1 \ge 1/\log q$, with $\eta = \eta(c)>0$ for any $c < 1/360$. From this, Theorem 1.2 asserts the averaged even moments of $S(t,f\otimes\chi)$ are $\frac{(2n)!}{n!(2\pi)^{2n}}(\log\log q)^n + O((\log\log q)^{n-1/2})$, and the normalized argument converges in distribution to the normal law with mean $0$ and variance $(2\pi^2)^{-1}$. The theorems are unconditional.

Load-bearing premise

The load-bearing premise is Lemma 2.4: the average over $\chi$ of $|L(s,f\otimes\chi)M(s,f\otimes\chi) - 1|^2$ is bounded by $(1+|t|)^A q^{-\eta(\sigma-1/2)}$ with a positive $\eta$, uniformly for $\sigma \ge 1/2 - 1/\log q$. If that exponent were $0$ or the uniformity failed near $\sigma = 1/2$, the zero-density theorem and the central limit theorem would lose their foundation.

Editorial extensions

If this is right

  • The distributional statement: as $q\to\infty$, $S(t,f\otimes\chi)/\sqrt{\log\log q}$ is normal with mean $0$ and variance $(2\pi^2)^{-1}$; the probability that it lies in $[a,b]$ tends to $\sqrt{\pi}\int_a^b \exp(-\pi^2\xi^2)\,d\xi$ uniformly in $a<b$.
  • The averaged even moments of $S(t,f\otimes\chi)$ are asymptotically those of a Gaussian: $\frac{(2n)!}{n!(2\pi)^{2n}}(\log\log q)^n$ with a relative error $O(1/\log\log q)$; hence $|S|$ is typically of size $\sqrt{\log\log q}$.
  • The zero-density estimate is non-trivial for $\sigma$ as close to $1/2$ as $1/\log q$, in windows of length as short as $1/\log q$, with a saving $q^{-\eta(\sigma-1/2)}$ uniform in the window's height.
  • Because the bound depends on $t_1,t_2$ only through the polynomial factor $(1+|t_1|+|t_2|)^A$, the estimate holds for zeros up to arbitrary height, matching the uniformity known for Dirichlet $L$-functions.

Reading between the lines

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

  • The paper's machinery is largely independent of the specific family: any $\mathrm{GL}_2$ or higher-rank family admitting a mollified second-moment bound of the shape $(1+|t|)^A q^{-\eta(\sigma-1/2)}$ would inherit a zero-density estimate and, with it, a moment formula and central limit theorem for the argument function — so the range of applicability is likely wider than twisted holomorphic forms
  • The restriction $c<1/360$ already suggests the argument works with very short mollifiers; if the exponent $\eta$ in the second-moment estimate could be made explicit, the method would yield a quantitative rate in the central limit theorem rather than the current qualitative convergence.
  • A testable extension: combining this character-aspect zero-density bound with an average over $t$ could give a joint $(t,\chi)$ normality statement for $S(t,f\otimes\chi)$, analogous to the two-parameter normality known for $\log\zeta$ — a direction the paper does not pursue.
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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 claims an unconditional Selberg-type zero-density estimate for the family of twisted GL2 L-functions L(s,f⊗χ), averaged over primitive Dirichlet characters χ modulo a prime q, and applies it to establish asymptotic formulas for the even moments of the argument function S(t,f⊗χ)=π^{-1}arg L(1/2+it,f⊗χ) and a central limit theorem. The proof follows Selberg's architecture: Lemma 2.4, imported from Blomer et al., supplies an averaged mollified second moment bound; Section 3.1 uses Selberg's argument principle to convert this into the zero-density estimate of Theorem 1.1; Section 4 approximates S(t,f⊗χ) by Dirichlet polynomials; Section 5 computes the moments of the main term and bounds the remainder. The paper is clearly organized and the overall strategy is standard, but the proof as written contains several gaps in load-bearing steps, particularly in the application of the argument principle, in a symmetry assertion about zeros, and in an orthogonality reduction.

Significance. If the gaps were repaired, Theorem 1.1 would be a valuable unconditional estimate for a family of GL2 L-functions for which few zero-density results are available, and Theorem 1.2 would be a natural GL2 analogue of Selberg's theorems on the distribution of S(t,χ). A notable strength is that the paper builds on a strong external input, the averaged second moment bound of Blomer, Fouvry, Kowalski, Michel, Milićević and Sawin, rather than on an unproved hypothesis. The paper is also well structured and the intended route is recognizable. For these reasons the central claims are worth pursuing, but the current manuscript does not yet justify them.

major comments (3)
  1. [§3.1, Lemma 2.3 and Remark 2] Lemma 2.3 is applied to ω(s)=1−(LM(s,f⊗χ)−1)^2 without verifying its main hypothesis. For fixed χ and t, as σ=Re s→∞, the absolutely convergent Dirichlet series give L=1+O(2^{-σ}) and M=1+O(2^{-σ}), hence ω(s)=1+O(4^{-σ}). Lemma 2.3 requires ω(s)=1+o(exp(−πRe s/(t2'−t1'))) = o(exp(−πσ log q/(1+2ϖ))). Since log 4≈1.386 is smaller than π log q/(1+2ϖ) for all sufficiently large q, the trivial pointwise decay 4^{-σ} is not o(exp(−πσ log q/(1+2ϖ))), so the hypothesis of Lemma 2.3 is not met. Remark 2 does not repair this: Lemma 2.4 is an averaged bound, and an average bound cannot imply a pointwise bound of size q^{-η(σ−1/2)/2}; at best it gives a pointwise bound of size q^{(1−η(σ−1/2))/2} after summing over the φ*(q) characters. The derivation of Theorem 1.1 therefore lacks a justified argument-principle step; a version of Lemma 2.3 that controls the averaged boundary contributions, or a finite-rectangle argument with a controlled error, is needed.
  2. [§4, Lemma 4.3] The proof of Lemma 4.3 asserts that if ρ=β+iγ is a zero of L(s,f⊗χ), then (1−β)+iγ is also a zero of L(s,f⊗χ). This is false for a general primitive complex character χ. The functional equation is Λ(s,f⊗χ)=ε Λ(1−s,f⊗\bar χ), so the reflected zero 1−ρ is a zero of L(s,f⊗\bar χ), not necessarily of L(s,f⊗χ). The lower-bound argument leading to equation (4.11), and hence the second part of Lemma 4.3 and Proposition 4.4, relies on this symmetry. Since Proposition 4.4 is used pointwise in the proof of (3.3), the remainder estimate is not established for non-real χ as written. An additional argument, for instance working with the pair χ and \bar χ or formulating the approximation on average over the family, is required.
  3. [§5.1, proof of (3.2)] The claim that “only the diagonal term m=n survives” under the assumption 0<δ<1/n is not justified. In (5.3), orthogonality forces p1⋯pm≡p_{m+1}⋯p_{2n} mod q, but when m≠n the two products need not be smaller than q: a product of up to 2n primes, each ≤q^δ, can exceed q even when δ<1/n. The congruence can therefore hold with m≠n without forcing equality of the two products. To make the stated diagonal reduction valid one would need the stronger condition δ<1/(2n), or one would need to bound the off-diagonal terms by a different argument such as Selberg's Lemma 2.2. This affects the main-term asymptotic formula (3.2) in Proposition 3.1.
minor comments (4)
  1. [§1, after (1.1)] The word “normalizd” should be “normalized”.
  2. [§3.1] The reduction to the case t2−t1=1/log q is asserted without explanation; the dyadic subdivision of longer intervals into short ones should be spelled out so that the dependence of the constants on the number of subintervals is visible.
  3. [Lemma 2.3] In the displayed statement of Lemma 2.3, the second integrand contains missing parentheses; it should read sinh(π(β−σ1)/(t2−t1)), and the convention for counting zeros on the boundary should be stated explicitly.
  4. [§5.1] After equation (5.1), the proof passes from a sum over all characters modulo q to the average over primitive characters without explicitly noting that the difference between φ(q) and φ*(q) is O(1) and harmless after division by φ*(q). This is a minor presentational point.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the central zero-density theorem is derived from an external second-moment theorem and Selberg's argument-principle lemma, and the moment/CLT applications use the derived theorem rather than assuming it.

full rationale

The paper's derivation chain is self-contained in the relevant sense. Theorem 1.1 is obtained in Section 3.1 by applying Selberg's argument-principle lemma (Lemma 2.3) to ω(s)=1−(LM−1)^2 and averaging over χ; the only estimate injected is Lemma 2.4, quoted from Blomer et al. [3, Theorem 8.5], which is external to the present authors and does not contain the target zero-count. The exponent η in Theorem 1.1 is inherited from Lemma 2.4 and the convergence condition η>π/(2ϖ+1) is imposed in the proof, not fitted to the conclusion. Theorem 1.2 then uses Theorem 1.1 through Lemma 5.1 to control the remainder R, while the main moment term in Proposition 3.1 comes from the diagonal contribution of ∑_p |λ_f(p)|^2/p (Lemma 2.1(3), Liu–Wang–Ye), a known result independent of the zero-density theorem. The self-citations [28,29,31] occur only in the introduction as background on prior GL2/GL3 results; none is load-bearing in the proofs. The skeptic objection that Lemma 2.3's uniform o(exp(−π Re(s)/(t2−t1))) hypothesis may fail for the chosen ω is a mathematical correctness risk, not a circularity: even if valid, it would invalidate the argument-principle step without making the conclusion identical to the input. No fitted parameter is relabeled as a prediction, and no definition is made in terms of the claimed outcome.

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

No numerical parameters are fitted to data. The proof parameters c, δ and ϖ are chosen small for convergence and do not carry empirical content. The central claim rests on standard results (Deligne bound, Selberg orthogonality, functional equation) and on the external Blomer et al. second moment theorem. No new entities or speculative objects are introduced.

assumptions (4)
  • standard math Deligne's proof of the Ramanujan-Petersson bound |λ_f(p)| ≤ τ(p)
    Used throughout to bound coefficients, for example in the error terms of Section 5.
  • standard math Selberg orthogonality for fixed holomorphic cusp form coefficients: ∑_{p≤x} |λ_f(p)|^2/p = log log x + O(1)
    Lemma 2.1(3), cited from Liu-Wang-Ye; it produces the log log q main term in the moment asymptotic.
  • domain assumption Blomer-Fouvry-Kowalski-Michel-Milicevic-Sawin mollified second moment theorem
    Lemma 2.4 is the external result that supplies the q^{-η(σ-1/2)} saving; the paper's zero-density theorem is built directly on it.
  • domain assumption The completed L-function Λ(s,f⊗χ) is entire and satisfies the standard functional equation with conductor q^2 r and gamma factor Γ(s+(k-1)/2)
    Invoked in Lemmas 2.3, 4.2 and the argument principle; standard for primitive twists of holomorphic newforms.

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Pith. "Pith review of A Selberg-type zero-density result for twisted $\rm GL_2$ $L$-functions and its application." pith.science (2026). https://pith.science/paper/SEU2P64J

@misc{pith2026250523573,
  author       = {Pith},
  title        = {Pith review of: A Selberg-type zero-density result for twisted $\rm GL_2$ $L$-functions and its application},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SEU2P64J}},
  note         = {Machine review of arXiv:2505.23573}
}
abstract

Let $f$ be a fixed holomorphic primitive cusp form of even weight $k$, level $r$ and trivial nebentypus $\chi_r$. Let $q$ be an odd prime with $(q,r)=1$ and let $\chi$ be a primitive Dirichlet character modulus $q$ with $\chi\neq\chi_r$. In this paper, we prove an unconditional Selberg-type zero-density estimate for the family of twisted $L$-functions $L(s, f \otimes \chi)$ in the critical strip. As an application, we establish an asymptotic formula for the even moments of the argument function $S(t, f \otimes \chi)=\pi^{-1}\arg L(1/2+\i t, f\otimes\chi)$ and prove a central limit theorem for its distribution over $\chi$ of modulus $q$.

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