Mean squares of quadratic twists of the Fourier coefficients of modular forms

Peng Gao; Yuetong Zhao

REVIEW 3 major objections 5 minor 15 references

Mean squares of quadratic twists of the Fourier coefficients of modular forms

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Theorem 1.1 supplies an unconditional asymptotic formula for a smoothed second moment of quadratic twists of Fourier coefficients of a fixed holomorphic Hecke eigenform…

desk verdict A credible, useful extension of the Soundararajan-Young method that needs one Euler-product verification before the main term is fully supported; worth refereeing. read the letter →

arxiv 2506.22256 v1 pith:VYUUIYYD submitted 2025-06-27 math.NT

Peng Gao , Yuetong Zhao This is my paper · ORCID

classification math.NT MSC 11N3711L0511L40

keywords meansquarequadraticDirichletcharactermodularL-functionstwistsFouriercoefficientsofcuspformssymmetricL-functionPoissonsummationasymptoticformula

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 establishes an unconditional asymptotic formula for a smoothed second moment of quadratic twists of the Fourier coefficients of a fixed holomorphic Hecke cusp form. Concretely, the weighted average over odd square-free $d$ of $|\sum_n \lambda_f(n)\chi_{8d}(n)\Phi(n/Y)|^2$, with an additional smooth weight $\Psi(d/X)$, equals a constant $C_0(\Phi,\Psi)$ times $XY$ plus explicitly bounded power-saving error terms, whenever $Y$ is a little smaller than $X$. The constant depends only on the test functions and on the symmetric-square $L$-function of $f$. This extends earlier asymptotic work on real character sums to sums twisted by modular-form coefficients, and it needs no unproved hypothesis such as the Riemann hypothesis. The main term emerges from the zero-frequency term of a Poisson summation, while the nonzero frequencies are controlled by a known second-moment bound for quadratic twists of modular $L$-functions.

What carries the argument

The central object is the Dirichlet series $$Z(u,v)=\sum_{\substack{(n_1n_2,2)=1\\ n_1n_2=\square}} \frac{\lambda_f(n_1)\lambda_f(n_2)}{n_1^u n_2^v}\prod_{p\mid n_1n_2}\frac{p}{p+1},$$ which arises from the $k=0$ term after Poisson summation and Möbius inversion. The proof uses the factorization quoted from [14, (4.6)], $$Z(u,v)=\zeta(u+v)L(2u,\mathrm{sym}^2 f)L(2v,\mathrm{sym}^2 f)L(u+v,\mathrm{sym}^2 f)Z_2(u,v),$$ with $Z_2$ absolutely convergent and uniformly bounded for $\Re u,\Re v>1/4$. This factorization is what allows the Mellin contour integrals to be shifted, producing the main term and the stated error. The nonzero frequencies are handled by decomposing the Gauss sums $G_k(n)=G_{4k}(n)$, writing $4k=k_1k_2^2$ with $k_1$ a fundamental discriminant, and applying the second-moment bound for quadratic twists of modular $L$-functions taken from [14, Corollary 2.5].

What would settle it

Check the factorization [14, (4.6)] directly at points with real parts just above $1/4$, say $u=v=0.3$, for an explicit eigenform such as the weight-12 discriminant form; the identity must hold with $Z_2$ bounded there. If the Dirichlet series defining $Z(u,v)$ fails to converge in that region, or if the factorization produces a pole or an unbounded factor there, the contour shifts that yield $C_0(\Phi,\Psi)XY$ and the error term would be invalid. As an end-to-end numerical check, one could also compute $S_f(X,Y;\Phi,\Psi)$ for a specific $f$ and smooth weights at a sequence of large $X,Y$ with $Y=X^{0.8}$ and examine whether $S_f/(XY)$ approaches the integral in (3.15).

Watch

Extended reading notes

Core claim

Theorem 1.1 is the central claim: for large $X,Y$ and any $\varepsilon>0$, $$S_f(X,Y;\Phi,\Psi)=C_0(\Phi,\Psi)XY+O\bigl($X^{{1/2+\varepsilon}}$$Y^{{3/2+\varepsilon}}$+$XY^{{1/2+\varepsilon}}$\bigr),$$ where $C_0(\Phi,\Psi)$ is the integral in (3.15), built from the Mellin transforms of the test functions, the values $L(2u,\mathrm{sym}^2 f)$, $L(2-2u,\mathrm{sym}^2 f)$, $L(1,\mathrm{sym}^2 f)$, and a bounded factor $Z_2(u,1-u)$. The proof is unconditional and works for any fixed holomorphic Hecke eigenform of weight divisible by 4. The square-free condition on $d$ is removed by Möbius inversion; the main contribution comes from the $k=0$ term in Poisson summation, and the nonzero frequencies are shown to contribute $Y^2Z$ after the optimization $Z=\sqrt{X/Y}$. The resulting asymptotic is valid whenever $Y\ll X^{1-\varepsilon}$, which is exactly the range where the displayed error terms are smaller than the main term.

Load-bearing premise

The load-bearing premise is that the quoted factorization [14, (4.6)] of the Dirichlet series $Z(u,v)$ is correct and that its remainder $Z_2(u,v)$ is uniformly bounded for $\Re u,\Re v>1/4$; if that identity or its stated region of convergence fails, the main term and all error estimates in Section 3.2 collapse.

Editorial extensions

If this is right

The smoothed second moment has a single constant-times-$XY$ main term, valid unconditionally for any $\varepsilon>0$ whenever $Y\ll X^{1-\varepsilon}$.
The final error terms in (1.7) come from balancing the Möbius-truncation error with the nonzero-frequency contribution, the balance being achieved at $Z=\sqrt{X/Y}$.
Together with the earlier conditional upper bounds quoted in the introduction, the theorem supports the uniform bound $S_m(X,Y;f)\ll XY^{m/2}(\log X)^{m(m-3)/2+1}$ for all real $m\ge 2$.
The proof transfers the Poisson-summation method, previously applied to real character sums, to sums weighted by modular-form coefficients, and it does so without assuming GRH.

Reading between the lines

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

The constant $C_0(\Phi,\Psi)$ in (3.15) is not fully explicit because it contains the bounded factor $Z_2$; obtaining numerical values for a specific form $f$ would require a separate evaluation of $Z_2$ along the line $v=1-u$.
The same contour-shift and factorization structure should plausibly extend to other $\mathrm{GL}(2)$ eigenforms or to higher moments, but the paper itself only claims the second-moment result for holomorphic Hecke eigenforms of weight divisible by 4.
The restriction $Y\ll X^{1-\varepsilon}$ leaves the diagonal regime $X\asymp Y$ open; sharper bounds on the nonzero-frequency contribution or on the second moment of quadratic twists could push the asymptotic range closer to that regime.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

This paper gives an unconditional asymptotic for a smoothed second moment of quadratic twists of the Fourier coefficients of a fixed modular form. That result is new: the earlier work by the first author handled the unweighted case, and the conditional bounds under GRH don't cover it. The main term is identified with a product of symmetric-square L-functions, and the error terms balance cleanly at Z = sqrt(X/Y). The argument follows the Soundararajan-Young template closely and is honest about doing so; there are no fitted parameters, and the reliance on [3] is not load-bearing. This is a credible, useful subfield result.

The main soft spot is the treatment of the Dirichlet series Z(u,v) after (3.13). The paper says it is essentially the function on [14, p. 1108] and then imports the factorization [14,(4.6)]. That factorization fixes the pole that produces the XY main term, and it drives the expression for C0. But the series here carries an extra Euler factor prod_{p|n1n2} p/(p+1) and the square-product condition n1n2 = □. The paper does not show that the local factors still match. I suspect they do, with the extra factor absorbed into the bounded Z2, but the proof should show it. As written, this is a load-bearing step that is asserted rather than verified. A referee should ask for that Euler-product computation, or at least a precise statement of the reduction to [14].

There is also a small misstatement: Theorem 1.1 says C0 depends only on Φ and Ψ, but (3.15) contains L(s, sym²f), so C0 depends on f as well. The dependence is harmless—f is fixed throughout—but the statement should be corrected.

The analytic continuation of Z is quoted from [14] without reproof. Given the closeness of the construction, I don't read that as a serious flaw, but it is part of the same under-verified region.

Overall, the skeleton is right, and I expect the missing local-factor check to go through. The paper deserves a serious referee and likely a conditional accept after a revision that fills that gap. It will be useful to people working on moments of quadratic character sums and families of L-functions. I'd bring it to a reading group if the group is in analytic number theory; otherwise it's a bit specialized.

Referee Report

3 major / 5 minor

Summary. The paper proves an unconditional asymptotic formula for a smoothed second moment of quadratic twists of the Fourier coefficients of a fixed holomorphic Hecke eigenform. The object is S_f(X,Y;Φ,Ψ) = ∑*_d Ψ(d/X)(∑_n λ_f(n)χ_{8d}(n)Φ(n/Y))². Theorem 1.1 claims S_f = C_0(Φ,Ψ)XY + O(X^{1/2+ε}Y^{3/2+ε} + XY^{1/2+ε}), with C_0 given by the integral in (3.15). The proof expands the square, applies Möbius inversion to remove the square-free condition on d, uses Poisson summation à la Soundararajan, isolates the k=0 term, and evaluates the resulting Dirichlet series Z(u,v) by quoting a factorization from Soundararajan–Young [14]. The remaining k≠0 terms are bounded after balancing the parameter Z = √(X/Y).

Significance. If correct, the result gives the first unconditional asymptotic in this range for the second moment of twisted Fourier-coefficient sums, extending the conditional moment bounds in [4] and the earlier smoothed result in [3]. The paper is concise, follows a well-established template, and has no fitted parameters; all displayed error terms do balance at the chosen value of Z. The main caveat is that the proof rests on an imported factorization whose exact match with the series Z(u,v) is not demonstrated, and one displayed normalization appears to contain a dimensional error. Because these points are fixable and the overall strategy is standard, the result is plausible but needs revision.

major comments (3)

[Section 3.2, after (3.13)] The identification of Z(u,v) with the function on [14, p. 1108] is load-bearing but is only asserted through the word 'essentially'. The series Z(u,v) contains the explicit local factor ∏_{p|n1n2} p/(p+1) and the condition n1n2 = □, and the quoted factorization [14, (4.6)] leads to the residue at v = 1-u and to the constant C_0. Please provide a direct local Euler-factor computation showing that this series equals ζ(u+v)L(2u,sym² f)L(2v,sym² f)L(u+v,sym² f)Z_2(u,v), and state precisely the region in which Z_2 is uniformly bounded and absolutely convergent. In particular, the claim that Z_2 converges absolutely for Re u, Re v > 1/4 is not immediate from the local factor, whose expansion contains a term of size p^{-(u+v)}; whether cancellation occurs should be shown explicitly. This point is central to the contour shifts in Section 3.2 and hence to the main term and the error estimate.
[Equation (3.11)] The definition of \tilde H_0(u,v) as ∫_0^∞ \hat h(xX; uY, vY) dx appears to introduce Y-dependent Mellin parameters in \hat h, which after the inversion in (3.11) does not reproduce the Mellin transform of H_0(n1,n2). The intended object should presumably be the Mellin transform of H_0 normalized independently of Y, e.g. Ψ̂(0)Φ̂(u)Φ̂(v), so that the factor Y^u Y^v provides the correct Y^{u+v}. As written, the powers of Y in the main term do not match the claimed XY, and the bound (3.12) is not consistent with a Y-dependent \tilde H_0. Please correct the normalization in (3.11) and verify that (3.12)–(3.15) remain unchanged.
[Theorem 1.1 and equation (3.15)] The constant C_0(Φ,Ψ) is stated to depend only on Φ and Ψ, but the integral in (3.15) contains L(2u,sym² f), L(2-2u,sym² f), L(1,sym² f), and Z_2(u,1-u), which depend on the fixed form f as well. Since f is fixed throughout, the wording is imprecise; it should read 'depending only on the fixed form f and on Φ, Ψ' or an equivalent phrase. The dependence on the auxiliary ε in the contour in (3.15) should also be addressed, since C_0 should be independent of the chosen line.

minor comments (5)

[Introduction, page 1] The phrase 'compacted supported functions' should read 'compactly supported functions'.
[Equation (3.7)] There is a duplicated article in 'the the O-term'; this should be corrected.
[Notation throughout Section 3] The parameter Z used for the Möbius cut in (3.1) and the Dirichlet series Z(u,v) introduced after (3.13) share the same symbol. This is confusing in a short argument; renaming one of them would improve readability.
[Theorem 1.1] The asymptotic formula is claimed for 'large X and Y', but the useful range is Y ≪ X^{1-ε}; this condition should be stated explicitly in the theorem or immediately after it, as is done in the discussion following (1.7).
[Equation (3.20)] The symbol CS in (3.16) and Lemma 3.4 is used for both the cosine/sine pair and the transform; the notation is acceptable but should be defined once in each occurrence to avoid ambiguity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the asymptotic formula is derived from external lemmas (Soundararajan [12], Soundararajan–Young [14]) with no fitted parameters and no use of the target result as an input.

full rationale

The proof of Theorem 1.1 is a direct application of externally established tools. The main term is produced by contour shifts applied to the Dirichlet series Z(u,v) in (3.13), whose analytic factorization is quoted from [14, (4.6)] rather than proved in the paper; this is an external mathematical input, not a restatement of Theorem 1.1, and it can be checked independently. The only appearances of the authors' own prior work are motivational (the discussion of [3] and [4] in the introduction); neither is used as a premise in the derivation of (1.7). No parameter is fitted to a subset of the target data and no quantity is renamed as a prediction. The paper's own equations do not identify the main-term constant C0(Φ,Ψ) with the input sum Sf by construction; C0 is a residue integral involving eH0 and known L-functions. Concerns about whether the 'essentially' identification of Z(u,v) with the function in [14] is fully justified are correctness or verification issues, not circularity. Accordingly no circular step is found.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters are fitted; the constant C0 is a closed-form integral of known L-functions. The proof is built on standard results from the literature (Soundararajan-Young [14], Soundararajan [12], Deligne, Shimura); no new entities are postulated.

assumptions (6)

domain assumption f is a fixed holomorphic Hecke eigenform of weight κ ≡ 0 (mod 4) for SL2(Z)
The setup of the theorem; weights divisible by 4 ensure χ8d is a primitive character for odd squarefree d.
standard math Deligne's Weil conjecture bound |λf(n)| ≤ d(n)
Used throughout to bound individual terms, e.g., in (3.8).
standard math Shimura's theorem on the analytic continuation and functional equation of L(s,sym² f)
Needed to justify contour shifts and convexity bounds in the main term, Section 3.2.
standard math The factorization of Z(α,β,γ;q,k1) in [14, Lemma 3.3] and the identity for Z(u,v) from [14, (4.6)]
This is the central structural identity that produces the main term; it is taken verbatim from the Soundararajan-Young paper.
standard math The second moment bound for quadratic twists of modular L-functions in [14, Corollary 2.5]
Applied to bound the tail terms in Lemma 3.1 and the k ≠ 0 contribution.
standard math Gauss sum evaluation Lemma 2.2 and Poisson summation formula Lemma 2.4 from Soundararajan's paper [12]
Used to transform the d-sum in (3.6).

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Cite this review

Pith. "Pith review of Mean squares of quadratic twists of the Fourier coefficients of modular forms." pith.science (2026). https://pith.science/paper/VYUUIYYD

@misc{pith2026250622256,
  author       = {Pith},
  title        = {Pith review of: Mean squares of quadratic twists of the Fourier coefficients of modular forms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VYUUIYYD}},
  note         = {Machine review of arXiv:2506.22256}
}

abstract

In this paper, we evaluate asymptotically a smoothed version of the sum \[ \displaystyle \sideset{}{^*}\sum_{d \leq X} \left( \sum_{n \leq Y} \lambda_f(n)\Big(\frac{8d}{n}\Big)\right)^2, \] where $\Big ( \frac{8d}{\cdot}\Big )$ is the Kronecker symbol, $\sideset{}{^*}\sum$ denotes a sum over positive odd square-free integers and $\lambda_f(n)$ are Fourier coefficients of a given modular form $f$.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

15 extracted references · 12 canonical work pages

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [3]

The mean square of real character sums

P. Gao, The mean square of real character sums (Preprint). arXiv:1908.07922

work page Pith review arXiv 1908

[2] [14]

Soundararajan and M

K. Soundararajan and M. P. Young, The second moment of quadratic twists of modular L-functions, J. Eur. Math. Soc. (JEMS) 12 (2010), no. 5, 1097–1116

work page 2010

[3] [4]

Bounds for Moments of Twisted Fourier coefficients of Modular Forms

P. Gao and L. Zhao, Bounds for moments of twisted Fourier coefficients of modular forms (Preprint). arXiv:2412.12515

work page Pith review arXiv

[4] [1]

M. V. Armon, Averages of real character sums , J. Number Theory 77 (1999), no. 2, 209–226

work page 1999

[5] [2]

Deligne, La conjecture de Weil

P. Deligne, La conjecture de Weil. I. , Inst. Hautes Etudes Sci. Publ. Math. 43 (1974), 273–307

work page 1974

[6] [5]

, Shifted moments of quadratic Dirichlet L-functions, Canad. Math. Bull. (to appear). arXiv:2406.18024

work page arXiv

[7] [6]

J. L. Hafner and A. Ivi´ c,On sums of Fourier coefficients of cusp forms , Enseign. Math. (2) 35 (1989), no. 3-4, 375–382

work page 1989

[8] [7]

A. J. Harper, Sharp conditional bounds for moments of the Riemann zeta function (Preprint). arXiv:1305.4618

arXiv