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arxiv: 2607.00282 · v1 · pith:AYQKRNY6new · submitted 2026-07-01 · 🧮 math.NT

Critical Zeros and Unconditional Mean Value Theorems for twisted hbox{PGL}(2) and hbox{PGL}(3) L-functions

Brian Conrey , Chung-Hang Kwan , Yongxiao Lin , Caroline L. Turnage-Butterbaugh This is my paper

Pith reviewed 2026-07-02 01:21 UTC · model grok-4.3

classification 🧮 math.NT MSC 11M2611F66
keywords L-functionscritical zerosLevinson methodmean value theoremsautomorphic representationsDirichlet charactersPGL(2)PGL(3)
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The pith

At least one ninth of the zeros of L(s, Π₀ × χ) lie on the critical line as the conductor Q tends to infinity.

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

The paper proves that in the family of L-functions attached to a fixed cuspidal automorphic representation Π₀ of PGL(3) twisted by primitive Dirichlet characters χ of conductor up to Q, at least 1/9 of the zeros lie on the critical line when Q is large. The argument uses Levinson's method and rests on a new mean-square asymptotic for L(s, Π₀ × χ) times an arbitrary Dirichlet polynomial that saves a positive power of Q uniformly in the range Q^ε ≤ T ≤ Q^{1/3-ε}. The result holds unconditionally when Π₀ is self-dual and under a mild extra hypothesis otherwise; the corresponding statements for PGL(2) are fully unconditional and quantitatively stronger. The same estimates supply evidence for the CFKRS conjectures at large height and large twist.

Core claim

As Q tends to infinity, at least 1/9 of the zeros of L(s, Π₀ × χ) lie on the critical line, where Π₀ is a cuspidal automorphic representation of PGL(3,A_Q) and χ runs over primitive Dirichlet characters of conductor ≤ Q. The result is unconditional for self-dual Π₀ and holds under a mild condition otherwise. For representations of PGL(2) the statements are fully unconditional and give a stronger proportion. The proof relies on a new power-saving asymptotic for the mean square of L(s, Π₀ × χ) times an arbitrary Dirichlet polynomial valid for T up to Q^{1/3-ε}.

What carries the argument

A new asymptotic formula with power-saving error term for the mean square of L(s, Π₀ × χ) times an arbitrary Dirichlet polynomial, valid uniformly in both the T- and Q-aspects for Q^ε ≤ T ≤ Q^{1/3-ε}.

If this is right

  • The same mean-value theorem yields the result for all self-dual Π₀ of PGL(3) without further hypotheses.
  • For PGL(2) the method produces a larger explicit proportion and applies without any extra conditions.
  • The estimates furnish evidence for the CFKRS conjectures when both the twist and the height are large.
  • The mean-value theorem is stated to be applicable to other problems in analytic number theory.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If similar mean-square asymptotics can be proved for higher-rank groups, the Levinson method would give positive proportions of critical zeros in those families as well.
  • Improving the error term in the mean square to allow longer Dirichlet polynomials might raise the 1/9 proportion.
  • The uniformity in both T and Q suggests the estimates could be used to study zeros near the edge of the critical strip in other twist families.

Load-bearing premise

The new mean-square asymptotic for the product of the twisted L-function and an arbitrary Dirichlet polynomial holds with a power-saving error term throughout the stated range of T and Q.

What would settle it

An explicit numerical check or theoretical construction showing that the proportion of critical zeros falls below 1/9 for some fixed self-dual Π₀ and a sequence of arbitrarily large Q would disprove the main claim.

read the original abstract

Let $\Pi_{0}$ be a cuspidal automorphic representation of $\mathrm{PGL}_{3}(\mathbb{A}_{\mathbb{Q}})$. In this paper, we use Levinson's method to prove that, as $Q\to \infty$, at least $1/9$ of the zeros of the $L$-functions $L(s, \Pi_{0}\,\times\, \chi)$ lie on the critical line, where $\chi$ ranges over the family of primitive Dirichlet characters of conductor up to $Q$. This result is unconditional when $\Pi_{0}$ is self-dual, and otherwise holds under a mild condition. The key technical input is a new asymptotic formula with a power-saving error term for the mean square of the product of $L(s, \Pi_{0}\times \chi)$ and a Dirichlet polynomial with arbitrary coefficients in both the $T$- and $Q$-aspects for the range $Q^{\epsilon}\le T \le Q^{1/3-\epsilon}$. When $T=Q^{\epsilon}$, our asymptotic formula allows Dirichlet polynomials of length $\theta <1/2-\epsilon$; when $\theta=0$, it gives a strong error term of size $O_{\epsilon}(Q^{7/4+\epsilon})$. Furthermore, our result provides evidence for the CFKRS conjectures for large twists and large vertical shifts. We also obtain corresponding results for $\mathrm{PGL}_{2}(\mathbb{A}_{\mathbb{Q}})$, which are fully unconditional, quantitatively stronger, and also appear to be new. This work develops a refined, flexible, and uniform version of the Asymptotic Large Sieve for $L$-functions that does not require any unproven progress toward the Generalized Ramanujan Conjecture. The arithmetic of $\Pi_{0}$ plays a crucial and delicate role in our argument. This work also makes extensive use of Mathematica to handle various elaborate Hecke algebra computations. Our mean value theorem is readily applicable to many other problems in analytic number theory.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The paper claims that as Q→∞, at least 1/9 of the zeros of L(s, Π₀ × χ) lie on the critical line, where Π₀ is a cuspidal automorphic representation of PGL(3, A_Q) and χ ranges over primitive Dirichlet characters of conductor ≤Q. The result is unconditional when Π₀ is self-dual (and holds under a mild condition otherwise). Analogous, quantitatively stronger results are obtained for PGL(2). The proof applies Levinson's method to a new asymptotic mean-square formula for L(s, Π₀ × χ) times an arbitrary Dirichlet polynomial, valid with power-saving error in the range Q^ε ≤ T ≤ Q^{1/3-ε} (with length restrictions when T=Q^ε). The work also develops a refined asymptotic large sieve without Ramanujan assumptions and provides evidence for CFKRS conjectures.

Significance. If the central claim holds, the result would constitute a notable advance by establishing the first unconditional positive-proportion theorems on critical zeros for twisted PGL(3) L-functions in large families, extending Levinson-type results beyond GL(2). The new mean-value theorem, uniform in both T and Q aspects and free of Ramanujan hypotheses, is a flexible technical tool with potential applicability to other analytic-number-theory problems. The explicit use of Hecke-algebra computations and the unconditional PGL(2) case are additional strengths.

major comments (1)
  1. [Abstract and key technical input paragraph] Abstract and paragraph on key technical input: the mean-square asymptotic is stated only for Q^ε ≤ T ≤ Q^{1/3-ε}. For fixed Π₀ on PGL(3) the analytic conductor is ≍ Q^3, so the total number of zeros with |Im ρ| ≪ Q^3 is ~ Q^3 log Q. The range T ≪ Q^{1/3} therefore controls only a Q^{-8/3+o(1)} fraction of all zeros. Levinson's method applied in this range yields at best a positive proportion among low-height zeros, which is o(1) of the full set; the manuscript must explain how the claimed 1/9 proportion of ALL zeros (without height restriction) follows.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The major comment raises an important point about the precise scope of our main theorem, which we address below by agreeing to revise the manuscript for clarity.

read point-by-point responses

  1. Referee: [Abstract and key technical input paragraph] Abstract and paragraph on key technical input: the mean-square asymptotic is stated only for Q^ε ≤ T ≤ Q^{1/3-ε}. For fixed Π₀ on PGL(3) the analytic conductor is ≍ Q^3, so the total number of zeros with |Im ρ| ≪ Q^3 is ~ Q^3 log Q. The range T ≪ Q^{1/3} therefore controls only a Q^{-8/3+o(1)} fraction of all zeros. Levinson's method applied in this range yields at best a positive proportion among low-height zeros, which is o(1) of the full set; the manuscript must explain how the claimed 1/9 proportion of ALL zeros (without height restriction) follows.

    Authors: We agree that the abstract and introduction as currently worded do not explicitly restrict attention to zeros of bounded height, which risks misinterpretation. Our Levinson-type argument, relying on the new mean-square asymptotic valid only for T ≪ Q^{1/3-ε}, in fact establishes that a positive proportion (at least 1/9) of the zeros lying in the range |Im ρ| ≪ Q^{1/3-ε} are on the critical line. This is still a new unconditional result for the PGL(3) family in the low-height regime. We will revise the abstract, the statement of the main theorem, and the discussion of the key technical input to make the height restriction explicit and to add a clarifying remark on the range of applicability. The PGL(2) results will be treated analogously where appropriate. revision: yes

Circularity Check

0 steps flagged

No circularity; new asymptotic supplies independent input to Levinson's method.

full rationale

The derivation proceeds by establishing a new power-saving asymptotic for the mean square of L(s, Π₀ × χ) times an arbitrary Dirichlet polynomial (valid for Q^ε ≤ T ≤ Q^{1/3-ε}), then feeding that formula into the standard Levinson mollifier argument to bound the proportion of zeros off the line. No equation is defined in terms of its own output, no fitted parameter is relabeled as a prediction, and no load-bearing step reduces to a self-citation whose content is itself unverified. The arithmetic of Π₀ and the Mathematica-assisted Hecke computations are external to the target proportion, so the chain remains self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper builds on standard axioms in number theory and introduces a new asymptotic formula as the key input. No free parameters or new entities are introduced in the abstract.

axioms (2)
  • standard math Properties of cuspidal automorphic representations and their L-functions
    Invoked in the definition of L(s, Π_{0} × χ) and the family.
  • domain assumption Levinson's method can be adapted to these twisted families
    Used to obtain the proportion of critical zeros from the mean value theorem.

pith-pipeline@v0.9.1-grok · 5937 in / 1560 out tokens · 50223 ms · 2026-07-02T01:21:35.453863+00:00 · methodology

discussion (0)

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Reference graph

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