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arxiv: 2410.20342 · v2 · submitted 2024-10-27 · 🧮 math.NT

On the second integral moment of L-functions

Liangxun Li This is my paper

Pith reviewed 2026-05-23 19:16 UTC · model grok-4.3

classification 🧮 math.NT
keywords automorphic L-functionsintegral momentsRamanujan conjectureGL(d)critical linenon-trivial boundssecond moment
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The pith

Assuming the generalized Ramanujan conjecture for GL_d L-functions with d at least 3, their second moment integral on the critical line saves a small power of log T over the size T to the d over 2.

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

The paper establishes that if the generalized Ramanujan conjecture holds for an automorphic L-function on GL_d over the rationals with d at least 3, then the integral from T to 2T of the square of its value at 1/2 plus it is bounded by T to the power d/2 divided by log to a small positive power eta_d. This gives a non-trivial improvement to the expected main term size of the moment. Readers focused on the size and distribution of L-functions would note that such bounds constrain how large these functions can be on average along the critical line. The argument applies once the conjecture is assumed and produces the saving for all such higher-rank cases.

Core claim

Assume that the generalized Ramanujan conjecture holds on the automorphic L-function L(s, π) on GL_d over Q with d≥3, we can obtain a small log-saving non-trivial bound on the second integral moment of L(1/2+it, π). Specifically the bound ∫_T^{2T} |L(1/2+it, π)|^2 dt ≪_π T^{d/2} / log^{η_d} T holds for a small constant η_d>0.

What carries the argument

The generalized Ramanujan conjecture on the local parameters of the automorphic representation, used to extract the logarithmic factor from the second moment integral.

If this is right

  • The second moment integral is bounded above by T^{d/2} divided by log to a positive power.
  • The bound is non-trivial relative to the main term size T^{d/2}.
  • The saving holds uniformly for every such L-function once the conjecture is granted.
  • The positive exponent η_d is small but depends only on the degree d.

Where Pith is reading between the lines

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

  • The conditional moment bound could be combined with other tools to obtain average-value results for these L-functions.
  • The approach links the Ramanujan conjecture directly to moment estimates that are often studied unconditionally only for lower rank cases.
  • If a larger saving were available under the same assumption it might affect subconvexity questions for higher rank L-functions.

Load-bearing premise

The generalized Ramanujan conjecture holds for the automorphic L-function on GL_d over Q with d at least 3.

What would settle it

An explicit automorphic representation π on GL_3 or higher rank for which the integral from T to 2T of |L(1/2+it, π)|^2 grows faster than T^{d/2} divided by any positive power of log T as T tends to infinity.

read the original abstract

Assume that the generalized Ramanujan conjecture holds on the automorphic $L$-function $L(s, \pi)$ on $\GL_d$ over $\mathbb{Q}$ with $d\geq 3$, we can obtain a small log-saving non-trivial bound on the second integral moment of $L(1/2+it, \pi)$. Specifically the bound \[ \int_{T}^{2T}\Big|L\big(\frac{1}{2}+it, \pi\big)\Big |^2 \dd t\ll_{\pi} \frac{T^{\frac{d}{2}}}{\log^{\eta_d}T} \] holds for a small constant $\eta_d>0$.

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

0 major / 3 minor

Summary. The manuscript assumes the generalized Ramanujan conjecture for the automorphic L-function L(s, π) attached to a cuspidal automorphic representation π on GL_d over Q with d ≥ 3. Under this hypothesis it derives a non-trivial upper bound for the second integral moment, specifically ∫_T^{2T} |L(1/2 + it, π)|^2 dt ≪_π T^{d/2} / log^{η_d} T for a small positive constant η_d that depends only on d.

Significance. Conditional on a standard conjecture, the result supplies a logarithmic saving over the convexity bound for the second moment of higher-rank L-functions. Such savings, even when small, are of interest in analytic number theory because they can sometimes be fed into applications involving subconvexity, zero-density estimates, or large-value estimates for L-functions.

minor comments (3)
  1. The abstract states the result clearly as conditional on GRC, but the manuscript should include an explicit statement of the precise form of GRC used (e.g., the Ramanujan bound on the Satake parameters) in the introduction or in a dedicated preliminary section.
  2. The dependence of η_d on d is described only as “small”; the paper should record the explicit value (or at least the order of magnitude) obtained from the argument so that readers can assess its utility.
  3. Notation for the implied constant ≪_π should be clarified: it is understood to depend on the fixed automorphic form π, but the paper should confirm that all other constants are absolute or depend only on d.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending minor revision. The referee's summary accurately reflects the main result: under the generalized Ramanujan conjecture, a small logarithmic saving is obtained for the second integral moment of L(1/2 + it, π) when d ≥ 3.

Circularity Check

0 steps flagged

No circularity; bound explicitly conditional on external GRC

full rationale

The paper states its main result as conditional on the generalized Ramanujan conjecture (a standard external hypothesis in automorphic forms, not derived or cited from the author's prior work). The abstract presents the assumption upfront and the resulting log-saving bound without any internal fitting, self-definition, or load-bearing self-citation. No equations or steps in the provided text reduce the claimed prediction to the input by construction. This is a standard conditional result with independent content under the stated hypothesis.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on one domain assumption (generalized Ramanujan conjecture) with no free parameters or invented entities visible in the abstract.

axioms (1)
  • domain assumption Generalized Ramanujan conjecture holds for automorphic L-functions on GL_d (d≥3)
    The bound is explicitly conditional on this conjecture as stated in the abstract.

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Reference graph

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15 extracted references · 15 canonical work pages · 1 internal anchor

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