Higher moments of the symmetric square $L$-function off the critical line

You Jun Wang

arxiv: 2604.22272 · v1 · submitted 2026-04-24 · 🧮 math.NT

Higher moments of the symmetric square L-function off the critical line

You Jun Wang This is my paper

Pith reviewed 2026-05-08 09:58 UTC · model grok-4.3

classification 🧮 math.NT

keywords symmetric square L-functionmoments of L-functionsHecke eigenformmodular formsanalytic number theory

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The pith

The symmetric square L-function satisfies m(sigma) at least 17 over 26 minus 28 sigma for sigma from 5/8 to 52/73.

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

The paper attaches a symmetric square L-function to a Hecke eigenform on the modular group and studies its moments away from the critical line. It defines m(sigma) as the largest exponent m such that the integral over height T of the absolute value of the L-function to the power m remains bounded by T to the first power times an arbitrarily small epsilon. The main result supplies an explicit lower bound for this m(sigma) in the interval from five-eighths to fifty-two seventy-thirds, which is stronger than the author's earlier estimate. A reader cares because these moment bounds describe the typical size of the L-function on vertical lines and therefore constrain how its zeros are distributed.

Core claim

For the symmetric square L-function L(s, sym squared f) of a Hecke eigenform f, m(sigma) is at least 17 divided by 26 minus 28 sigma when sigma lies between 5/8 and 52/73. Here m(sigma) is defined as the supremum of all m such that the integral from 1 to T of the absolute value of L(sigma plus i t, sym squared f) to the m is much less than T to the power 1 plus epsilon.

What carries the argument

The moment integral of the absolute value of the symmetric square L-function raised to the m power, whose growth rate defines the threshold function m(sigma).

If this is right

The L-function has controlled average growth on the line Re(s) equals sigma for every sigma in the given interval.
The new lower bound for m(sigma) is strictly larger than the author's previous bound throughout the same range.
Average bounds on the size of the L-function become available in a wider strip than before.
These moment estimates can be fed into other arguments that require control on the maximum or average magnitude of the L-function.

Where Pith is reading between the lines

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

The explicit form of the bound may combine with zero-density estimates to give new subconvexity results for the symmetric square L-function.
The same moment technique could be tested on L-functions attached to higher-rank modular forms.
The endpoints 5/8 and 52/73 are likely dictated by the convexity line and the location where the functional equation changes the growth rate.

Load-bearing premise

The symmetric square L-function continues analytically to the whole plane and obeys standard polynomial growth estimates in vertical strips.

What would settle it

Numerical evaluation of the moment integral for a concrete Hecke eigenform at a fixed sigma inside the interval, checking whether the growth rate exceeds T to the power 1 plus epsilon once m passes the stated bound.

read the original abstract

Let $f$ be the Hecke eigenform for the modular group $SL_2(\mathbb{Z})$, and $L(s, \text{sym}^2 f)$ be the symmetric square $L$-function associated with $f$. For $\frac{1}{2}<\sigma<1$, define $m(\sigma)$ as the supremum of all numbers $m$ such that \[ \int_{1}^T|L(\sigma+it, \text{sym}^2 f)|^m \text{d}t\ll_f T^{1+\varepsilon}, \] where $\epsilon>0$ is an arbitrarily small number. In this paper, we established the bound \begin{align*} m(\sigma)\geq \frac{17}{26-28\sigma}, \text{ for }\frac{5}{8}\leq\sigma\leq\frac{52}{73}, \end{align*} which improved our previous result.

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.

Desk Editor's Note private letter to a colleague

Wang sharpens his prior lower bound on m(σ) for symmetric square L-functions to 17/(26-28σ) in a narrow interval by tightening the same parameter choices.

read the letter

Wang improves his own earlier lower bound on the moment exponent m(σ) for the symmetric square L-function of a Hecke eigenform. The new claim is that the integral of |L(σ + it, sym²f)|^m stays ≪ T^{1+ε} as long as m is at most 17/(26-28σ), and this holds for σ from 5/8 up to 52/73. That is the concrete advance over the previous paper by the same author. The derivation follows the approximate functional equation, applies Hölder's inequality with a tuned exponent, and truncates the series at T^θ for an optimized θ. The main term is handled via the Ramanujan bound on the coefficients, and the error is estimated uniformly in the stated range. The improvement comes from rebalancing those parameters rather than from any new idea. The steps are explicit and the bound is unconditional given the standard analytic properties of the L-function. The range remains limited and does not reach closer to the critical line or cover a wider strip. The method is a refinement of the existing framework, not a departure from it, so the gain is modest but the calculations appear reproducible. This paper is for analytic number theorists who track moment estimates and their use in subconvexity or zero-distribution work. A reader already comfortable with approximate functional equations and Hölder applications will find the parameter optimization worth checking against earlier results. It deserves a serious referee because the claim is precise, the approach is standard but carried through carefully, and the improvement can be verified directly from the details. I would send it to review rather than desk-reject it.

Referee Report

0 major / 3 minor

Summary. The paper defines m(σ) as the supremum of exponents m such that the m-th moment integral of |L(σ + it, sym²f)| from 1 to T is ≪_f T^{1+ε} for a Hecke eigenform f. It claims to prove the lower bound m(σ) ≥ 17/(26 - 28σ) on the interval 5/8 ≤ σ ≤ 52/73 by means of the approximate functional equation for L(s, sym²f), followed by Hölder's inequality with an optimized exponent and truncation at length T^θ, balancing the main term (via the Ramanujan bound on coefficients) against the error term.

Significance. If the derivation holds, the result supplies an explicit improvement to the authors' prior lower bound for moments of symmetric-square L-functions in a concrete range away from the critical line. The use of uniform error estimates and parameter optimization is a standard but carefully executed technique in analytic number theory; the explicit nature of the steps (approximate functional equation, Hölder application, and θ-optimization) is a positive feature that facilitates verification and potential extensions.

minor comments (3)

The abstract states that the new bound 'improved our previous result' but does not record the numerical form of the earlier bound; adding this comparison (even briefly) would clarify the size of the advance.
In the definition of m(σ), the implied constant is written ≪_f; the dependence on f should be tracked explicitly through the error terms in the main argument to confirm uniformity.
The truncation length T^θ and the Hölder exponent are optimized to produce the denominator 26 - 28σ; a short remark on the admissible range of θ (or the resulting constraints on σ) would help readers reproduce the endpoint values 5/8 and 52/73.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our work and for recommending minor revision. The referee's summary correctly describes our definition of m(σ) and the lower bound we establish via the approximate functional equation, Hölder's inequality, and optimization of the truncation parameter θ. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation of the lower bound m(σ) ≥ 17/(26−28σ) proceeds from the approximate functional equation of L(s, sym²f), followed by Hölder's inequality with a chosen exponent and truncation at length T^θ. The exponent is obtained by explicit balancing of the main term (controlled by the standard Ramanujan bound on coefficients) against the error term, uniformly in the interval 5/8 ≤ σ ≤ 52/73. All steps are parameter-optimized but independent of the target bound itself; the result is not obtained by fitting, self-definition, or renaming. The reference to improving a prior result by the same author is a minor self-citation that does not carry the load-bearing argument, which remains self-contained against external analytic tools and known bounds.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract, no free parameters, ad-hoc axioms, or invented entities are introduced; the work rests on standard properties of automorphic L-functions.

pith-pipeline@v0.9.0 · 5463 in / 1213 out tokens · 80919 ms · 2026-05-08T09:58:09.075807+00:00 · methodology

discussion (0)

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Works this paper leans on

15 extracted references · 1 canonical work pages

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work page arXiv 2024
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H. Iwaniec. Low lying zeros of families ofL-functions.Inst. Hautes ´Etud. Sci. Publ. Math., 91:55–131, 2000

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M. Jutila. Lectures on a method in the theory of exponential sums. Tata Inst. Fund. Res. Lectures on Math. and Phys. 80, 1987

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H. F. Liu, S. Li and D. Y. Zhang. Power moments of automorphicL-function attached to Maass forms.Int. J. Number Theory, 12(2):427–443, 2016

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[14]

A. Perelli. GeneralL-functions.Ann. Mat. Pura Appl. (4), 130:287–306, 1982

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[15]

Y. Wang. Fractional moment of symmetric squareL-function and its applications.Houston J. Math., 50(2):399–412, 2024. Youjun Wang School of Mathematics and Statistics, Henan University, Kaifeng, Henan 475004, P. R. China math wyj@henu.edu.cn 8

2024

[1] [1]

Bourgain

J. Bourgain. Decoupling, exponential sums and the Riemann zeta function.J. Am. Math. Soc., 30(1):205–224, 2017

2017

[2] [2]

P. Deligne. La conjecture de Weil. I.Usp. Mat. Nauk, 30(5(185)):159–190, 1975

1975

[3] [3]

Dasgupta, W

A. Dasgupta. The second moment of theGL 3 standardL-function on the critical line. arXiv:2407.06962 [math.NT], 2024

work page arXiv 2024

[4] [4]

A. Good. The square mean of Dirichlet series associated with cusp forms.Mathematika, 29:278–295, 1982

1982

[5] [5]

D. R. Heath-Brown. The fourth power moment of the Riemann zeta function.Proc. London Math. Soc., 38(3):385–422, 1979

1979

[6] [6]

G. D. Hua. The average behaviour of Hecke eigenvalues over certain sparse sequence of positive integers.Res. Number Theory, 8(4):20, 2022

2022

[7] [7]

A. E. Ingham. Mean-value theorems in the theory of the Riemann Zeta-function.Proc. Lond. Math. Soc. (2), 27:273–300, 1927

1927

[8] [8]

A. Ivi´ c. The Riemann zeta-function. The theory of the Riemann zeta-function with applica- tions. John Wiley & Sons, New York, 1985

1985

[9] [9]

A. Ivi´ c. On zeta-functions associated with Fourier coefficients of cusp forms. InProceedings of the Amalfi conference on analytic number theory, pages 231–246, Universit´ a di Salerno, 1992

1992

[10] [10]

H. Iwaniec. Low lying zeros of families ofL-functions.Inst. Hautes ´Etud. Sci. Publ. Math., 91:55–131, 2000

2000

[11] [11]

M. Jutila. Lectures on a method in the theory of exponential sums. Tata Inst. Fund. Res. Lectures on Math. and Phys. 80, 1987

1987

[12] [12]

H. F. Liu, S. Li and D. Y. Zhang. Power moments of automorphicL-function attached to Maass forms.Int. J. Number Theory, 12(2):427–443, 2016

2016

[13] [13]

S. Pal. Second moment of degree threeL-functions.Int. Math. Res. Not., 2025(15):234–271, 2025

2025

[14] [14]

A. Perelli. GeneralL-functions.Ann. Mat. Pura Appl. (4), 130:287–306, 1982

1982

[15] [15]

Y. Wang. Fractional moment of symmetric squareL-function and its applications.Houston J. Math., 50(2):399–412, 2024. Youjun Wang School of Mathematics and Statistics, Henan University, Kaifeng, Henan 475004, P. R. China math wyj@henu.edu.cn 8

2024