REVIEW 4 major objections 7 minor 2 cited by
Bounds for Moments of Twisted Fourier coefficients of Modular Forms
T0 review · 4 major / 7 minor · reviewed 2026-08-11 · deepseek-v4-flash
Pith's one-line read Under GRH, twisted modular coefficient sums obey character-sum moment bounds.
desk verdict A solid GRH-conditional extension of the Soundararajan–Harper method to twisted modular-form coefficients, but Theorem 1.5 rests on an unproved key estimate that a referee should demand be written out. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing tool is a shifted-moment estimate for the twisted modular $L$-functions (Theorems 1.1 and 1.2): for a product $\prod_j |L(1/2+it_j,f\otimes\chi)|^{a_j}$ summed over characters, the bound is $\phi(q)(\log q)^{(a_1^2+\cdots+a_k^2)/4}$ times products of $\zeta(1+i(t_j-t_l)+1/\log q)$ and $L(1+i(t_j-t_l)+1/\log q,\mathrm{sym}^2 f)$ with exponents $a_ja_l/2$, and an analogous formula holds for quadratic twists. Corollaries 1.3 and 1.4 compress these factors into the piecewise functions $g_1$ and $g_2$, which are then summed over dyadic regions. For the coefficient moments, the paper smooths the sum with a bump function, applies Mellin inversion to write it as an integral of $L(1/2+it,f\otimes\chi)$, and bounds the $2m$-th power of that integral by convexity and dyadic decompositions. In the fixed-modulus case the argument relies on the estimate (4.12) for $\int_0^B |L(1/2+it,f\otimes\chi)|\,dt$, which the paper says follows by modifying a cited proposition without supplying the proof.
What would settle it
Compute $S_3(q,Y;f)$ for a fixed eigenform such as the discriminant cusp form $\Delta$, over a range of large $q$ and $Y\le q$, and compare with $\phi(q)Y^3(\log q)^4$: any excess power of $\log q$ would refute Theorem 1.5. More directly, the unproved estimate (4.12) can be checked numerically for $m=3$ and several $q$, since the paper's fixed-modulus theorem collapses if that integral-moment bound is false.
Extended reading notes
Core claim
The central claim is that character-twisted sums of modular-form Fourier coefficients inherit the moment bounds known for character sums. Concretely, Theorem 1.5 gives $S_m(q,Y;f)\ll\phi(q)Y^m(\log q)^{(m-1)^2}$ for real $m>2$ and $Y\le q$, and Theorem 1.6 gives $T_m(X,Y;f)\ll XY^m(\log X)^{2m^2-3m+2}$ for $m\ge2$ and $Y\le X$, both under GRH. The exponent of the logarithm is the substantive part: the $Y^m$ factor is the trivial diagonal size, while the log powers encode the distribution of primes weighted by $\lambda_f(p)$ and $\lambda_f(p^2)$. The paper obtains these via new shifted-moment bounds for $L(1/2+it,f\otimes\chi)$, in which the dependence on the shifts is expressed through nearby values of $\zeta(s)$ and $L(s,\mathrm{sym}^2 f)$.
Load-bearing premise
The load-bearing premise is that the unproved estimate (4.12), controlling the integral moment $\int_0^B |L(1/2+it,f\otimes\chi)|\,dt$, truly follows by a straightforward modification of the cited proposition, and that GRH holds for the twisted $L$-functions and their symmetric squares.
Editorial extensions
If this is right
- Standard interpolation extends the statements to every real $m>0$: $S_m(q,Y;f)\ll\phi(q)Y^m(\log q)^{O(1)}$ and $T_m(X,Y;f)\ll XY^m(\log X)^{O(1)}$.
- Setting $k=1$ in Theorem 1.6 recovers the clean quadratic-twist bound $T_m(X,Y;f)\ll XY^m(\log X)^{2m^2-3m+2}$ for all $m\ge2$.
- The shifted-moment theorems give a reusable template for bounding moments of twisted modular $L$-functions at nearby points, with shift differences entering through $\zeta$ and the symmetric-square $L$-function.
- Under GRH the $2m$-th moments are as small as the diagonal contribution allows: essentially $\phi(q)Y^m$ or $XY^m$ up to powers of logarithms.
Reading between the lines
- The same shifted-moment machinery should carry over to other fixed Hecke eigenforms, higher levels, or other automorphic families whenever the symmetric-square input is available; this is an extension, not a paper claim.
- The gap around estimate (4.12) means the fixed-modulus result is conditional on an unproved modification; a reader building on this paper should first supply a full proof of that integral-moment estimate.
- The bound shapes suggest matching lower bounds: it is natural to conjecture that the exponents $(m-1)^2$ and $2m^2-3m+2$ are best possible under GRH, paralleling the known sharpness for character sums.
- A numerical check of $S_3(q,Y;f)$ for modest $q$ and $Y$ would test the predicted $\log q$ exponent; the paper does not report such data.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. This paper works under GRH for the twisted L-functions and their symmetric squares. Theorem 1.1 bounds shifted moments over primitive characters chi modulo q of products |L(1/2+it_j, f⊗chi)|^{a_j} by phi(q)(log q)^{sum a_j^2/4} times products of zeta and symmetric-square factors at 1+i(t_j-t_l)+1/log q; Theorem 1.2 gives the analogous bound for quadratic twists chi(8d) over odd squarefree d≤X, with zeta and L(s,sym^2 f) factors at t_j±t_l and 2t_j. Corollaries 1.3 and 1.4 replace these factors by the majorants g1, g2. These estimates are then applied, following Szabó [12] and the authors' preceding work [4], to bound moments of character-twisted sums of the Fourier coefficients lambda_f(n): Theorem 1.5 gives S_m(q,Y;f) ≪ phi(q)Y^m(log q)^{(m-1)^2} for real m>2 and Y≤q, and Theorem 1.6 gives T_m(X,Y;f) ≪ X Y^m(log X)^{2m^2-3m+2} for m≥2 and Y≤X. The proofs use the Soundararajan-Harper method: pointwise majorants for log|L| (Section 2), Harper-type dissections into exceptional sets (Section 3), and dyadic decomposition of the t-integrals (Sections 4-5). The quadratic-side integral-moment bound, Proposition 5.4, is proved in detail; the corresponding estimate on the Dirichlet side, (4.12), is not.
Significance. If fully proved, the results are a natural and worthwhile extension: Theorem 1.5 reproduces for Dirichlet twists of a modular form the leading shape of Szabó's character-sum bounds, and Theorem 1.6 extends the authors' quadratic-Dirichlet-L-function work [4] to quadratic twists of modular L-functions. The statements are consistent with the expected optimal exponents under GRH, and the use of the parameter k in Proposition 5.4 is a sensible device that keeps the dyadic sums convergent. Credit is due for writing out the quadratic-twist integral-moment argument in real detail, for cleanly separating zeta and symmetric-square contributions via g1 and g2, and for stating the GRH hypotheses precisely. The manuscript is also transparent about which passages are abbreviated. However, the proofs of Theorems 1.1, 1.2, and 1.5 are not complete at precisely the points where the text says 'straightforward modification' or 'similar to [12, Section 3]': the estimate (4.12) and the passage to (4.13) carry the full weight of Theorem 1.5, and identity (3.6) together with Lemma 2.8 carries the full weight of Theorem 1.2.
major comments (4)
- [§4.4, Eqs. (4.12)–(4.13)] Lemma 4.2 is the only input for Theorem 1.5, and it rests entirely on the estimate (4.12), which is asserted to follow from 'a straightforward modification' of the proof of [12, Proposition 3]. This cannot be accepted as printed for several reasons. First, Corollary 1.3 is stated for a fixed integer k and positive exponents a_j, while (4.12) is needed for a real exponent 2m with m>2; the interpolation argument, or the k-variable plus extra-integral device later used in Proposition 5.4, is not given. Second, the stated justification that the functions g2 from (1.4) are bounded by (log log B)^{O(1)} is only valid when B < eq; (4.12) is stated for B = q^{O(1)}, and for B ≥ eq Corollary 1.3 gives g2(x) = log q, so the reason supplied does not cover the stated range. Third, the transition from (4.12) to (4.13) is not shown: in a dyadic decomposition the second term of (4.12) carries a factor e^{2mn} that cancels the e^{-2mn} weight, leaving a sum over n ≤ ε log q that requires an additional saving not reproduced from [12, Section 3]; without that saving the exponent (log q)^{(m-1)^2} is not obtained for all m>2. Finally, (4.13) is dimensionally inconsistent: its left-hand side is independent of Y while the right-hand side contains Y^m, and inserting (4.13) into (4.11), which already contains a factor Y^m, would produce Y^{2m} in the conclusion of Lemma 4.2 rather than the claimed Y^m. The proof of Theorem 1.5 is therefore incomplete as printed.
- [§3, around Eq. (3.6); Lemma 2.8] Identity (3.6) is the core of the proof of Theorem 1.2, and it is introduced with the sentence 'We proceed by a straightforward modification of the proof of [4, Theorem 1.1] upon using Lemma 2.8'. The omitted material includes: the definition and use of the exceptional sets S(j); the high-moment bounds for aℜM_{m,l}(d) that give these sets small measure; the control of the tail of the prime sum beyond the largest scale; and the conversion from the smoothed sum over d of |A(d)L|^{a_1}...|A(d)L|^{a_k} back to the unweighted moment, which requires Lemma 2.8 with the real weight A(d)^{-a}, where a = sum_j a_j. Lemma 2.8 itself is stated as 'a straightforward modification of the proof of [4, Lemma 2.4]', but it involves a real exponent k in the factor A(d)^{-k} and an error term O_k(X^{1/2+ε}n^{1/4+ε}) whose size, after summation over the tuples of primes arising in the moment expansion, is not discussed. Since Corollary 1.4, Proposition 5.4, and Theorem 1.6 all depend on this chain, the quadratic side of the paper also lacks a complete derivation at its load-bearing point.
- [§3, Eq. (3.7)] The displayed 'direct computation' in (3.7) contains a sign error in its second line. Using h(p) = ½∑_m a_m p^{-it_m}, one has (2ℜh(p))²/(2p) − ℜh(p²)/p = (1/(2p))[½∑_j a_j² + ∑_{i<j} a_i a_j (cos((t_i+t_j)log p)+cos((t_i−t_j)log p)) + ∑_j (a_j²/2 − a_j) cos(2t_j log p)], so the coefficient of cos(2t_j log p) should be (a_j²/2 − a_j), not (a_j²/2 + a_j) as printed. With the printed sign, (3.7) contradicts the paper's own Theorem 1.2 and Corollary 1.4, where the zeta-factor exponent at 1+2it_j is a_j²/4 − a_j/2. The final statements are recovered from the corrected computation, but the proof as displayed is internally inconsistent; the sign must be fixed and the surrounding steps re-verified.
- [§3, Eqs. (3.1)–(3.3)] The proof of Theorem 1.1 is also a compressed reference ('Proceeding as in the proof of [12, Section 4]'). A concrete point that needs explication is the following: with the definition (3.1), all scales satisfy α_i ≤ 10^{−M} for i ≤ J, so the dissection treats only primes p ≤ q^{10^{−M}}, yet the bound (3.3) contains the full sum exp(∑_{p≤q}|h(p)λ_f(p)|²/p) over primes up to q. The treatment of the primes in (q^{10^{−M}}, q] — one of the central steps of Harper's method — is not described anywhere in Section 3. As Theorem 1.1 and Corollary 1.3 feed directly into the unproved estimate (4.12) flagged in Major Comment 1, this omission is part of the same load-bearing gap.
minor comments (7)
- [§4.4, Eq. (4.12)] The symbol y in the factor (log log y)^{O(1)} on the right-hand side of (4.12) is not defined; the intended scale (presumably Y or the dyadic block e^n) should be stated.
- [§5.3, Eq. (5.5)] In the inequality preceding (5.6), the exponent of j should be 2m−2k (consistent with the following line and with (5.6)), not 2m−k as printed.
- [§2.1, Eq. (2.6)] The displayed definition of Λ(s, sym² f ⊗ χ) has L(s, sym² f) on the right-hand side; it should be L(s, sym² f ⊗ χ), as the surrounding text and (2.5) make clear.
- [§2.2, Eq. (2.34)] In Proposition 2.10, the term (1+λ)(log q + log(|t|+2)) should presumably read (1+λ)(log q + log(|t|+2))/log x; as printed the bound is inconsistent with its use in Corollary 2.11, equation (2.43), where the corresponding term is 2(A+1)log q/log x.
- [§4.5, Lemma 4.3] Lemma 4.3 is stated for all m ≥ 1, but its proof applies (4.12) with exponent 4m−2, which the hypothesis m>2 of (4.12) covers only when m>1; the case m=1 would require a separate second-moment estimate, although this case is not needed for Theorem 1.5.
- [§2.9, Lemmas 2.15–2.16] Lemmas 2.15 and 2.16 are asserted to follow by modifying [12, Proposition 1] and [5, Theorem 2] without proof; since both are used at σ up to 1+1/log Y (e.g., in (4.23) and (5.15)), a sentence on the uniformity in σ of the implied (log q)^{O(1)} bounds would be helpful.
- [§3, Eq. (3.7)] The second displayed line of (3.7) contains a duplicated summation symbol '∑_{p≤X}∑_{p≤X}'; one of the two should be removed.
Circularity Check
No circularity: the central moment bounds are derived from separately established shifted-moment theorems, not from the target sums; cited prior work is used as method, not as an equivalent input.
full rationale
The paper's derivation chain is non-circular. Theorems 1.5 and 1.6 are deduced from the shifted moment bounds in Theorems 1.1 and 1.2, which are proved in Section 3 by Harper's method starting from the pointwise GRH estimates in Corollaries 2.13 and 2.14. The target quantities S_m and T_m are character sums over n≤Y; they are not used to define the L-function moments that enter the proof, and no fitted parameter is renamed as a prediction. Self-citations to [4] are methodological analogues (for example, Lemma 2.8 is said to follow by modifying [4, Lemma 2.4]) and are not used to import the conclusion of [4] in place of a proof. The genuinely fragile step is estimate (4.12) in the proof of Lemma 4.2: it is asserted to follow by a 'straightforward modification' of [12, Proposition 3], and the stated bound on the functions g2 by (log log B)^{O(1)} is not justified for the full range B=q^{O(1)}; moreover (4.13) appears to carry an extra Y^m on the right. These are completeness or correctness concerns, not instances of a claim reducing to its own inputs. No self-definitional, fitted-input, uniqueness-import, ansatz-smuggling, or renaming pattern is present. I therefore find no circularity.
Assumptions & free parameters
assumptions (4)
- domain assumption GRH for L(s, f⊗χ), L(s, sym² f⊗χ), and L(s, χ) for the relevant characters
- domain assumption f is a fixed holomorphic Hecke eigenform of weight κ ≡ 0 (mod 4) for SL2(Z)
- standard math Deligne's proof of the Weil conjectures gives |α_p| = |β_p| = 1 and α_p β_p = 1
- standard math Known analytic properties of ζ(s), L(s, sym² f), and twisted symmetric square L-functions, including continuation, functional equations, and no pole at s = 1
Cite this review
Pith. "Pith review of Bounds for Moments of Twisted Fourier coefficients of Modular Forms." pith.science (2026). https://pith.science/paper/JECMZWIX
@misc{pith2026241212515,
author = {Pith},
title = {Pith review of: Bounds for Moments of Twisted Fourier coefficients of Modular Forms},
year = {2026},
howpublished = {\url{https://pith.science/paper/JECMZWIX}},
note = {Machine review of arXiv:2412.12515}
}
abstract
We establish upper bounds for shifted moments of modular $L$-functions to a fixed modulus as well as quadratic twists of modular $L$-functions under the generalized Riemann hypothesis. Our results are then used to establish bounds for moments of sums involving with Fourier coefficients of a given modular form twisted by Dirichlet characters.
Forward citations
Cited by 2 Pith papers
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Mean squares of quadratic twists of the Fourier coefficients of modular forms
An unconditional asymptotic formula is obtained for the smoothed mean square of quadratic twists of Fourier coefficients of a modular form.
-
Bounds for moments of twisted quadratic characters of prime modulus
For prime moduli p, the smoothed m-th moment of quadratic twists of a fixed modular form's coefficients grows like X Y^{m/2} (log X)^{m(m-3)/2}, with matching even-m lower bounds, conditional on GRH.
Reference graph
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