Pith. sign in

REVIEW 2 major objections 1 minor 16 references

The $\theta = \infty$ Conjecture and the Riemann Hypothesis for Automorphic $L$-functions

T0 review · 2 major / 1 minor · reviewed 2026-06-30 · grok-4.3

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Bounded mollified second moments for arbitrarily long mollifiers imply non-vanishing regions for GL_m automorphic L-functions.

desk verdict The paper shows that the θ=∞ moment conjecture for GL_m L-functions implies non-vanishing in strips and a family-level quasi-RH, by extending Bettin-Gonek. read the letter →

arxiv 2605.24363 v1 pith:GF2IU4MC submitted 2026-05-23 math.NT

Anji Dong , Nawapan Wattanawanichkul , Alexandru Zaharescu This is my paper · ORCID
classification math.NT
keywords automorphicL-functionsmollifiedmomentsnon-vanishingRiemannhypothesisGL_mthetainfinityconjecturecriticalstrip
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 develops an analogue of the θ=∞ conjecture for L-functions attached to cuspidal automorphic representations on GL_m. It proves that if the mollified second moments remain bounded when the mollifier has arbitrary polynomial length, then the L-functions have no zeros in corresponding regions of the critical strip. The same moment condition, when assumed for an entire family, yields a quasi-Riemann hypothesis for that family. The argument extends the Bettin-Gonek method from the zeta function to these higher-rank L-functions.

What carries the argument

The mollified second moment of the L-function taken against a mollifier of arbitrary polynomial length, whose boundedness is shown to control zero locations via an extension of the Bettin-Gonek contour integration and mean-value estimates.

What would settle it

An explicit calculation demonstrating that the mollified second moment exceeds any fixed bound for some sequence of polynomial-length mollifiers applied to a concrete GL_2 or GL_3 L-function, or the location of a zero lying outside the predicted non-vanishing region for that L-function.

Watch

Extended reading notes

Core claim

Extending the Bettin-Gonek framework, the authors prove that suitable bounds on the mollified second moments of GL_m automorphic L-functions, holding for mollifiers of arbitrary polynomial length, imply that these L-functions have no zeros in corresponding regions inside the critical strip. They further show that the θ=∞ conjecture for a family of such L-functions implies a quasi-Riemann hypothesis for the family.

Load-bearing premise

The analytic framework of Bettin and Gonek extends to automorphic L-functions on GL_m without new obstructions that would prevent the non-vanishing conclusion from following from the moment bound.

Editorial extensions

If this is right

  • Individual GL_m L-functions satisfy explicit zero-free regions inside the critical strip whenever their mollified second moments remain bounded for long mollifiers.
  • A family satisfying the θ=∞ conjecture has all but a zero-density set of zeros lying in a narrow vertical strip around the critical line.
  • The non-vanishing criterion applies uniformly to the family once the moment bound is verified for the family as a whole.
  • The length of the mollifier directly determines the width of the zero-free region obtained.

Reading between the lines

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

  • Numerical checks of the moment bound for low-degree cases such as elliptic-curve L-functions could provide early evidence for or against the conjecture.
  • If the moment condition can be verified for short mollifiers and then extended, it would give a practical route to partial zero-free regions.
  • The same technique might connect to other families where moment asymptotics are already known, transferring those results into quasi-Riemann hypotheses.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 1 minor

Summary. The paper introduces an analogue of the θ=∞ conjecture for cuspidal automorphic L-functions on GL_m and proves that if the mollified second moments remain bounded for mollifiers of arbitrary polynomial length, then the L-functions are non-vanishing in corresponding regions of the critical strip. It further shows that the family version of the conjecture implies a quasi-Riemann Hypothesis for the family, by extending the Bettin-Gonek analytic framework.

Significance. If the extension of the Bettin-Gonek machinery holds without obstruction, the result supplies a conditional route from moment bounds to zero-free regions for higher-rank L-functions, which is of interest for understanding zero distributions beyond the zeta function. The conditional character of the implication is stated clearly and the family version adds a useful generalization.

major comments (2)
  1. [Abstract] Abstract (paragraph on extending the framework): the claim that the Bettin-Gonek approximate functional equation, off-diagonal estimates, and contour-shifting argument extend to GL_m without new obstructions from the product of m Gamma factors or conductor growth is asserted but not verified in detail; for m>1 the shifts in the Gamma factors alter the support of the mollifier integral and could change the size of the error terms that must be controlled to pass from the moment bound to the non-vanishing statement.
  2. [Abstract] The family version of the criterion (final paragraph of the abstract): the passage from the θ=∞ conjecture for a family to a quasi-RH requires uniform control over the family of the off-diagonal terms after the extension; no explicit uniformity statement or dependence on the family parameters is supplied in the abstract, which is load-bearing for the quasi-RH conclusion.
minor comments (1)
  1. Notation for the completed L-function and the precise form of the mollifier should be introduced earlier to make the extension statements easier to follow.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the constructive major comments. We respond to each point below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [Abstract] Abstract (paragraph on extending the framework): the claim that the Bettin-Gonek approximate functional equation, off-diagonal estimates, and contour-shifting argument extend to GL_m without new obstructions from the product of m Gamma factors or conductor growth is asserted but not verified in detail; for m>1 the shifts in the Gamma factors alter the support of the mollifier integral and could change the size of the error terms that must be controlled to pass from the moment bound to the non-vanishing statement.

    Authors: The body of the manuscript (Sections 3 and 4) carries out the extension explicitly, adapting the approximate functional equation, deriving the off-diagonal estimates, and performing the contour shifts while tracking the m-fold Gamma product and conductor growth. The error terms are bounded in terms of the assumed mollified-moment hypothesis, and the support of the mollifier integral is adjusted accordingly. Nevertheless, we agree that a more self-contained verification of the m>1 case would strengthen the presentation; we will add a short subsection that isolates the differences from the m=1 case and confirms that the same error-term controls suffice. revision: yes

  2. Referee: [Abstract] The family version of the criterion (final paragraph of the abstract): the passage from the θ=∞ conjecture for a family to a quasi-RH requires uniform control over the family of the off-diagonal terms after the extension; no explicit uniformity statement or dependence on the family parameters is supplied in the abstract, which is load-bearing for the quasi-RH conclusion.

    Authors: The full paper states the family theorem with explicit uniformity hypotheses on the off-diagonal terms (uniform in the family parameters under the stated growth conditions). The abstract is intentionally concise, but we accept that a brief reference to this uniformity would clarify the load-bearing step. We will revise the final sentence of the abstract to note that the quasi-RH follows under uniform control of the off-diagonal contributions. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: conditional implication from external moment conjecture

full rationale

The derivation establishes an implication (bounded mollified second moments for arbitrary polynomial-length mollifiers imply non-vanishing in corresponding regions of the critical strip, and the family version yields quasi-RH). This is presented as extending the Bettin-Gonek framework without reducing the target non-vanishing statement to a definition, fit, or self-citation chain inside the paper. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear in the abstract or described claims; the central result remains an independent conditional statement anchored outside the paper's own inputs.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The central claim rests on the validity of extending the Bettin-Gonek moment-to-zero machinery to GL_m L-functions; no explicit free parameters or invented entities are mentioned in the abstract.

assumptions (1)
  • standard math Standard analytic properties of automorphic L-functions on GL_m, including functional equations and Euler products
    Invoked when extending the Bettin-Gonek framework to these L-functions (abstract).

how reviews work

0 comments
Cite this review

Pith. "Pith review of The $\theta = \infty$ Conjecture and the Riemann Hypothesis for Automorphic $L$-functions." pith.science (2026). https://pith.science/paper/GF2IU4MC

@misc{pith2026260524363,
  author       = {Pith},
  title        = {Pith review of: The $\theta = \infty$ Conjecture and the Riemann Hypothesis for Automorphic $L$-functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GF2IU4MC}},
  note         = {Machine review of arXiv:2605.24363}
}
abstract

The $\theta=\infty$ conjecture asserts that the mollified second moments of the Riemann zeta function remain bounded for mollifiers of arbitrary polynomial length. We investigate an analogue of this conjecture for automorphic $L$-functions associated with cuspidal representations of $\text{GL}_m(\mathbb{A}_{\mathbb{Q}})$, exploring its implications for the distribution of their nontrivial zeros. Extending the framework of Bettin and Gonek, we prove that if the mollified second moments of these $L$-functions remain suitably bounded for mollifiers of arbitrary polynomial length, then the $L$-functions are non-vanishing in corresponding regions of the critical strip. Furthermore, we establish a version of this criterion for families of $L$-functions, demonstrating that the $\theta = \infty$ conjecture for a family of $L$-functions implies a quasi-Riemann Hypothesis for that family.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

  1. [1]

    Bettin and S

    S. Bettin and S. M. Gonek. Theθ=∞conjecture implies the Riemann Hypothesis.Mathematika, 63(1):29–33, 2017

    work page 2017

  2. [2]

    J. B. Conrey. More than two fifths of the zeros of the Riemann zeta function are on the critical line.J. Reine Angew. Math., 399:1–26, 1989

    work page 1989

  3. [3]

    J. B. Conrey, H. Iwaniec, and K. Soundararajan. Critical zeros of DirichletL-functions.J. Reine Angew. Math., 681:175–198, 2013

    work page 2013

  4. [4]

    M. K. Das and S. Pujahari. Mean of the product of derivatives of Hardy’sZ-function with Dirichlet polynomial.J. Number Theory, 258:334–367, 2024

    work page 2024

  5. [5]

    D. W. Farmer. Long mollifiers of the Riemann zeta-function.Mathematika, 40(1):71–87, 1993

    work page 1993

  6. [6]

    S. Feng. Zeros of the Riemann zeta function on the critical line.J. Number Theory, 132(4):511–542, 2012

    work page 2012

  7. [7]

    Iwaniec and P

    H. Iwaniec and P. Sarnak. Perspectives on the analytic theory ofL-functions.Geom. Funct. Anal., Special Volume, Part II:705–741, 2000. GAFA 2000 (Tel Aviv, 1999)

    work page 2000

  8. [8]

    G. Ji. Lower bounds for moments of automorphicL-functions over short intervals.Proc. Amer. Math. Soc., 137(11):3569–3574, 2009

    work page 2009

Show all 16 references
  1. [9]

    K¨ uhn, N

    P. K¨ uhn, N. Robles, and D. Zeindler. On a mollifier of the perturbed Riemann zeta-function.J. Number Theory, 174:274–321, 2017

    2017

  2. [10]

    Levinson

    N. Levinson. More than one third of zeros of Riemann’s zeta-function are onσ= 1/2.Advances in Math., 13:383–436, 1974

    1974

  3. [11]

    W. Luo, Z. Rudnick, and P. Sarnak. On Selberg’s eigenvalue conjecture.Geom. Funct. Anal., 5:387–401, 1995

    1995

  4. [12]

    M¨ uller and B

    W. M¨ uller and B. Speh. Absolute convergence of the spectral side of the Arthur trace formula for GL n. Geom. Funct. Anal., 14(1):58–93, 2004

    2004

  5. [13]

    Pratt, N

    K. Pratt, N. Robles, A. Zaharescu, and D. Zeindler. More than five-twelfths of the zeros ofζare on the critical line.Res. Math. Sci., 7(2):Paper No. 2, 74, 2020

    2020

  6. [14]

    Radziwi l l

    M. Radziwi l l. Limitations to mollifyingζ(s).Preprint, arXiv:math.NT/1207.6583., 2012

    2012 arXiv

  7. [15]

    Robles, A

    N. Robles, A. Roy, and A. Zaharescu. Twisted second moments of the Riemann zeta-function and appli- cations.J. Math. Anal. Appl., 434(1):271–314, 2016

    2016

  8. [16]

    X. Wu. The twisted mean square and critical zeros of DirichletL-functions.Math. Z., 293(1-2):825–865, 2019. Anji Dong: Department of Mathematics, University of Illinois Urbana-Champaign, Alt- geld hall, 1409 W. Green Street, Urbana, IL, 61801, USA Email address:anjid2@illinois...

    2019

Pith tools

Reviewed June 30, 2026 · model on record in the stance chip above.