REVIEW 3 major objections 4 minor 29 references
Bounds for moments of twisted quadratic characters of prime modulus
T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash
Pith's one-line read Under GRH, for every integer $m\ge 4$ the smoothed $m$-th moment of the sums twisted by $\chi_{8p}$ over primes $p\le X$ is $O(XY^{m/2}(\log X)^{m(m-3)/2})$, and for even $m$ this order is matched from below.
desk verdict Solid prime-modulus analogue of known moment bounds, but the main upper bound rests on an estimate cited from the authors' own unpublished preprint, and the abstract oversells the unsmoothed case. 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 object is the family of shifted moments of twisted modular $L$-functions, namely $\sum_{2<p\le X}(\log p)\prod_{j}|L(1/2+it_j,f\otimes\chi_{8p})|^{a_j}$, bounded in Theorem 2.1 and converted in Proposition 2.8 into explicit factors $g_1,g_2$ depending on the sizes of $|t_i\pm t_j|$. The proof of that shifted-moment bound follows the standard conditional-moment route: a GRH-conditional upper bound on $\log|L(1/2+it,f\otimes\chi_{8p})|$ (Lemma 2.7) is combined with a dyadic decomposition of the primes according to the size of short Dirichlet polynomials, a technique developed in the study of high moments of $\theta$ functions. The lower bound isolates the diagonal contribution $n_1\cdots n_m=\square$, whose size is computed by Lemma 2.6 via Mellin inversion and the product formula $G(s)=\prod_j L(2s_j,\mathrm{sym}^2 f)\prod_{l_1<l_2}\zeta(s_{l_1}+s_{l_2})L(s_{l_1}+s_{l_2},\mathrm{sym}^2 f)E(s)$. The sharp upper bound for integer moments is then reduced, following the approach used for quadratic character sums, to the integral estimate $I_{m,B,k}$ quoted as (5.3).
What would settle it
Compute the integral $I_{4,B,1}$ from (5.2)–(5.3) for a fixed large $B$ and an increasing sequence of $X$; if it grows faster than $C B^2(\log X)^2(\log\log B)^C$ for every fixed $C$, then Theorem 1.2's logarithmic exponent is false. A direct check would be to compute $U_4(X,Y;f,W)$ for a small form such as the weight-$12$ cusp form and moderate $X,Y$; growth beyond $XY^2(\log X)^{2+\delta}$ for some $\delta>0$ would contradict the claimed upper bound.
Extended reading notes
Core claim
The paper's central claim is that, under GRH, the true order of the smoothed prime-modulus moment is $XY^{m/2}(\log X)^{m(m-3)/2}$ up to constants: Theorem 1.2 gives the upper bound for all integers $m\ge 4$, and Theorem 1.3 gives the matching lower bound for all even integers $m\ge 4$ in the range $X^\varepsilon\ll Y\ll X^{1-\alpha}$. The exponent $m(m-3)/2$ comes from the $\binom{m}{2}$ pair factors $\zeta(s_{l_1}+s_{l_2})L(s_{l_1}+s_{l_2},\mathrm{sym}^2 f)$ in the Dirichlet series of the diagonal condition $n_1\cdots n_m=\square$, so the lower bound isolates exactly that diagonal main term. For the unsmoothed moment the paper establishes the softer bound $U_m(X,Y;f)\ll XY^{m/2}(\log X)^{O_{m,\varepsilon}(1)}$ for every real $m>0$, which fixes the main term $XY^{m/2}$ up to a bounded power of $\log X$.
Load-bearing premise
Beyond the Generalized Riemann Hypothesis itself, the sharp upper bound rests on an estimate for the integral $I_{m,B,k}$ that the paper cites from the authors' own earlier preprint [14] and does not prove here.
Editorial extensions
If this is right
- For even integers $m\ge 4$, Theorems 1.2 and 1.3 fix the exact order $XY^{m/2}(\log X)^{m(m-3)/2}$ of the smoothed prime-modulus moment under GRH.
- The bound removes the extra $(\log X)^2$ that the previous general estimate (Theorem 1.1 with $k=1$) left in the exponent, matching the sharp exponent already known for the squarefree-modulus family.
- For every real $m>0$, the unsmoothed moment satisfies $U_m(X,Y;f)\ll XY^{m/2}(\log X)^{O_{m,\varepsilon}(1)}$, so the main term $XY^{m/2}$ is correct up to a power of $\log X$ depending only on $m$.
- The lower bound shows that for even moments the diagonal condition $n_1\cdots n_m=\square$ alone determines the order of the moment; any further main terms would only affect the leading constant, not the order.
- Via standard moment inequalities, these bounds translate into distributional control on the character sums $\sum_{n\le Y}\chi_{8p}(n)\lambda_f(n)$ as $p$ varies among primes up to $X$.
Reading between the lines
- The proof of Theorem 1.2 reduces the sharp upper bound to the unproved integral estimate (5.3) from the authors' earlier preprint [14]; proving or disproving that estimate would settle the smoothed upper bound independently of the rest of the argument.
- The same diagonal-plus-shifted-moment strategy should extend to prime-modulus moments of quadratic Hecke $L$-functions over number fields, provided an analogue of Theorem 2.1 holds in that setting.
- For odd integers $m\ge 5$ there is no matching lower bound here; if the exponent $m(m-3)/2$ still holds, the diagonal alone cannot be the whole story, so odd moments would require a genuinely non-diagonal mechanism.
- A numerical test for $m=4$ over moderate ranges of $X$ and $Y$ could check whether the ratio $U_4(X,Y;f,W)/(XY^2(\log X)^2)$ stabilizes, which would be evidence for a leading constant; the paper does not compute such constants.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies, under GRH, the moments U_m(X,Y;f) (and a smoothed variant U_m(X,Y;f,W)) of sums of Fourier coefficients of a fixed holomorphic Hecke eigenform twisted by the quadratic character χ_{8p}, summed over odd primes p≤X. Theorem 1.1 gives an upper bound for real m≥4 with a logarithmic exponent R(m,k,ε) by splitting into a smooth part and an unsmoothed part; the proof is carried out in Sections 3–4. Theorem 1.2 claims the sharp smoothed upper bound U_m(X,Y;f,W) ≪ X Y^{m/2} (log X)^{m(m−3)/2} for integers m≥4. Theorem 1.3 gives a matching smoothed lower bound for even integers m≥4 in the range X^ε ≪ Y ≪ X^{1−α}. The main new input for the prime-modulus family is Theorem 2.1, a shifted-moment bound for L(1/2+it_j, f⊗χ_{8p}), proved in Section 3, and its consequence Proposition 2.8. The proof of Theorem 1.2, however, reduces at the decisive point to the integral estimate (5.3) taken from the authors' own preprint [14], and the proof of Theorem 1.3 relies on Lemma 2.6, whose proof is also borrowed from [14].
Significance. If all imported estimates are valid, the paper would establish the exact order of the smoothed even-integer moments in the prime-modulus quadratic-twist family and would confirm the conjectured logarithmic exponent m(m−3)/2 in (1.2) for that family. The paper also contains a substantial, essentially self-contained proof of the real-m upper bound in Theorem 1.1, including the shifted-moment theorem (Theorem 2.1), which is genuinely new for prime moduli and is proved in some detail in Section 3. Credit is due for the detailed treatment of Theorem 2.1 and Proposition 2.8, and for the lower-bound argument in Theorem 1.3, which uses Lemma 2.4 and Lemma 2.6 in a clean way. The main caveat is that the sharp upper bound, the paper's headline result, is not derived in this manuscript: it is imported from the authors' preprint [14] via (5.3), and the proof of the key combinatorial estimate Lemma 2.6 is likewise delegated to [14]. Thus the marginal contribution over [14] is real but is more incremental than the abstract suggests.
major comments (3)
- [Section 5, Eq. (5.3)] The proof of Theorem 1.2 is not self-contained at its decisive step. After reducing U_m(X,Y;f,W) to the bound (5.1) for Z_m(B,X), the paper defines the integral I_{m,B,k} and asserts, in (5.3), that I_{m,B,k} ≪ B^2 (log X)^{m(m−3)/2} (log log B)^{O_m(1)}, with the sentence that this was shown in [14] during the proof of [14, Theorem 1.1]. No proof or even a sketch of (5.3) appears here. Since [14] is the authors' own unpublished preprint, the sharp logarithmic exponent in Theorem 1.2 rests entirely on an external, not-yet-available argument. If (5.3) is off by one power of log X, the claimed exponent in (1.6) does not follow from this paper; and Theorem 1.3 would then be a lower bound for a different smoothed object rather than the matching upper bound. I request that a full proof of (5.3), or at least a complete outline, be included, or that Theorem 1.2 be restated explicitly as conditional on [14, Theorem 1.1] with a clear indication of which parts are new.
- [Abstract vs. Theorems 1.1–1.3] The abstract states that the paper establishes 'the correct order of magnitude for the unsmoothed m-th moment for all real m≥4'. This is not what the theorems show. Theorem 1.1 gives only the upper bound U_m(X,Y;f) ≪ X Y^{m/2} (log X)^{O_{m,ε}(1)}; no unsmoothed lower bound is proved for any real m. Theorem 1.3 is a lower bound only for the smoothed moment U_m(X,Y;f,W), only for even integers m≥4, and only in the range X^ε ≪ Y ≪ X^{1−α}. The abstract should be revised to describe exactly these results, e.g., a sharp smoothed upper bound for integer m≥4, a matching smoothed lower bound for even m≥4, and a conditional upper bound of the unsmoothed moment for real m with a weaker logarithmic exponent.
- [Section 2, Lemma 2.6 and Section 3, Lemma 3.1] Two lemmas that are load-bearing for the two theorems are asserted without proof. Lemma 2.6, which gives P_f(Y^{β_1},…,Y^{β_m}) ≍ Y^{Σβ_i/2} (log Y)^{m(m−3)/2}, is used directly in the lower-bound proof of Theorem 1.3 (Eq. (6.4) and (6.6)); its proof is dismissed with 'the argument follows directly from the proof of [14, Lemma 2.5]'. Lemma 3.1, which controls the contribution of the set S(0) in the proof of Theorem 2.1, is stated with the comment 'The proof is almost the same as that of [29, Lemma 3.1], so we omit it.' For a journal submission, these are not acceptable substitute for proofs, especially because the whole point of the paper is to extend the methods of [14] and [29] to the prime-modulus case. Please supply the missing arguments or state the results as quoted theorems from the cited sources with full statements.
minor comments (4)
- [Various] There are several typographical slips: 'Theoem 2.1' in the paragraph after Lemma 2.7, and the display in Lemma 2.6 and later in (6.4)–(6.6) writes 'Y β_i' instead of 'Y^{β_i}'.
- [Section 4.1, Eq. (4.24)] In the proof of Proposition 4.2, the parameter D is introduced as a general parameter in (4.1), then later set to D=X^{3ε}; the reader would benefit from an explicit sentence saying that this choice is made at that point and is admissible.
- [References] Reference [14] is an arXiv preprint by the same authors and is cited for the central estimate (5.3); since that estimate is load-bearing, the manuscript should indicate whether [14] has been accepted for publication, and if not, should include the proof here.
- [Section 5, Eq. (5.2)] The derivation of the second inequality in (5.2), where g_2 is estimated trivially using (2.20), is terse; a sentence clarifying why the (log log B)^{O_m(1)} factor absorbs the relevant contributions would improve readability.
Circularity Check
The sharp (log X)^{m(m-3)/2} exponent in both Theorems 1.2 and 1.3 is imported from the authors' own preprint [14], not derived in this paper.
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self citation load bearing
[Section 5, proof of Theorem 1.2, equation (5.3)]
"The estimate for I_{m,B,k} was obtained in the course of the proof of [14, Theorem 1.1]. Specifically, it was shown in [14] that, for all integers m≥4 and 1≤k≤3, I_{m,B,k} ≪ B^2 (log X)^{m(m-3)/2} (log log B)^{O_m(1)}. (5.3) The desired bound in (5.1) then follows directly from (5.2) and (5.3). This completes the proof of Theorem 1.2."
Theorem 1.2 reduces U_m(X,Y;f,W) to Z_m(B,X), then via (5.2) to I_{m,B,k}, and then stops at (5.3), citing the authors' own preprint [14] for the decisive sharp exponent. No proof or sketch of (5.3) is given here, and [14] is neither machine-checked nor otherwise independently verified in this paper. If (5.3) were off by even one power of log X, the claimed upper bound would not follow. This is load-bearing self-citation for the paper's central upper-bound claim.
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self citation load bearing
[Section 2.3, proof of Lemma 2.6; used in Theorem 1.3 via (6.4)-(6.6)]
"As these product representations are identical in form to those in [14, Lemma 2.4], and the remainder factors E(s), E+(s) retain good convergence properties in Re(s_j)>1/4, the argument follows directly from the proof of [14, Lemma 2.5]."
Lemma 2.6 supplies the exact lower-bound exponent by asserting P_f(Y^{β1},...,Y^{βm}) ≍ Y^{Σβ_i/2}(log Y)^{m(m-3)/2}. Its proof is entirely delegated to [14, Lemmas 2.4-2.5], again the same authors' preprint. Theorem 1.3 uses this lemma through equations (6.4)-(6.6) to bound H_2 and H_1, so the matching lower bound's sharp logarithm also rests on the self-cited [14] rather than on an argument contained in this paper.
full rationale
The paper does contain genuine independent work: Section 3 proves the prime-modulus shifted moment theorem (Theorem 2.1) via the Soundararajan-Harper method, and Section 4 proves Theorem 1.1, a looser unsmoothed upper bound. Those portions are not circular. However, the two refined results advertised as the main achievements are not self-contained at the decisive step. The proof of Theorem 1.2 ends by importing estimate (5.3) from [14], the authors' own preprint, with no proof or stated verifiable assumptions, and the proof of Theorem 1.3 depends on Lemma 2.6, whose proof is explicitly declared to follow directly from [14, Lemma 2.5]. Thus the sharp logarithmic exponent in both the upper and lower bounds is inherited from the same self-cited source. This is not identity-by-construction and no fitted parameter is disguised as a prediction, so the score is not 9 or 10; but because the central claim itself reduces to load-bearing self-citation, a score of 6 is appropriate. A secondary, non-circular issue is that the abstract claims the correct order of magnitude for the unsmoothed m-th moment for all real m≥4, whereas Theorem 1.1 provides only an upper bound with a larger exponent and Theorem 1.3 is a smoothed even-integer lower bound; no unsmoothed lower bound is proved.
Assumptions & free parameters
assumptions (4)
- domain assumption GRH for L(s, f tensor chi_{8p}) for all odd primes p, and the RH/GRH-conditional estimates in Lemma 2.2 and Lemma 2.4.
- ad hoc to paper The integral estimate (5.3): I_{m,B,k} << B^2 (log X)^{m(m-3)/2} (log log B)^{O_m(1)}, asserted to be proved in [14, Theorem 1.1].
- ad hoc to paper Lemma 2.5 and Lemma 2.6: the Euler product decompositions for G(s) and G_+(s), and the size estimate P_f(Y^{beta}) asymptotically equal to Y^{sum beta_i/2} (log Y)^{m(m-3)/2}, adapted from [14, Lemmas 2.4 and 2.5].
- standard math Standard background: Deligne's bound |lambda_f(n)| <= d(n), holomorphy and functional equation of L(s, sym^2 f) (Shimura), Heath-Brown's mean value estimate [18, Corollary 2], and the Soundararajan-Harper shifted moment machinery [27, 17].
Cite this review
Pith. "Pith review of Bounds for moments of twisted quadratic characters of prime modulus." pith.science (2026). https://pith.science/paper/I6SGJRKT
@misc{pith2026260805961,
author = {Pith},
title = {Pith review of: Bounds for moments of twisted quadratic characters of prime modulus},
year = {2026},
howpublished = {\url{https://pith.science/paper/I6SGJRKT}},
note = {Machine review of arXiv:2608.05961}
}
abstract
We study, under the Generalized Riemann Hypothesis (GRH), the moments of sums of Fourier coefficients of a fixed holomorphic Hecke eigenform twisted by the quadratic character $\chi_{8p}$, where $p$ ranges over odd primes. We establish the correct order of magnitude for the unsmoothed $m$-th moment for all real $m\geq 4$, and a sharp upper bound of order $XY^{m/2}\,\, (\log X)^{m(m-3)/2}\,\,$ for the smoothed $m$-th moment for all integers $m\geq 4$. A matching lower bound for all even integers $m\geq 4$ shows that this bound is optimal.
Reference graph
Works this paper leans on
- [14]
-
[25]
M. Munsch and Y. Toma. Bounds for moments of quadratic character sums and theta functions.Bull. Sci. Math., 210:Paper No. 103806, 31, 2026
work page 2026
-
[29]
Y. Zhao. Bounds for moments of quadratic dirichlet character sums of prime moduli.Indagationes Mathe- maticae, 37(4):1557–1585, 2026. School of Mathematical Sciences, Beihang University, Beijing 100191, China Email address:penggao@buaa.edu.cn School of Mathematical Sciences, China University of Geosciences (Beijing), Beijing 100191, China Email address:yu...
work page 2026
-
[1]
J. C. Andrade and S. Baluyot. Small zeros of DirichletL-functions of quadratic characters of prime modulus. Res. Number Theory, 6(2):1–20, 2020
work page 2020
-
[2]
J. C. Andrade and J. P. Keating. Mean value theorems forL-functions over prime polynomials for the rational function field. Preprint. arXiv:1401.0418
-
[3]
M. V. Armon. Averages of real character sums.J. Number Theory, 77(2):209–226, 1999
work page 1999
-
[4]
Dirichlet $L$-functions of quadratic characters of prime conductor at the central point
S. Baluyot and K. Pratt. DirichletL-functions of quadratic characters of prime conductor at the central point. Preprint. arXiv:1809.09992
-
[5]
H. M. Bui and A. Florea. Zeros of quadratic DirichletL-functions in the hyperelliptic ensemble.Trans. Amer. Math. Soc., 370(11):8013–8045, 2018
work page 2018
Show all 29 references
-
[6]
P. Deligne. La conjecture de Weil. I.Inst. Hautes Etudes Sci. Publ. Math., 43:273–307, 1974
1974
-
[7]
P. Gao, X. He, and X. Wu. Bounds for moments of modularL-functions to a fixed modulus.Acta Arith., 205(2):161–189, 2022
2022
-
[8]
Gao and L
P. Gao and L. Zhao. One level density of low-lying zeros of quadratic heckeL-functions to prime moduli. Hardy-Ramanujan Journal, 43:173–187, 2021
2021
-
[9]
Gao and L
P. Gao and L. Zhao. Bounds for moments of quadratic dirichletL-functions of prime-related moduli.Colloq. Math., 171:61–77, 2023
2023
-
[10]
Gao and L
P. Gao and L. Zhao. Bounds for moments of quadratic Dirichlet character sums.Bull. Aust. Math. Soc., 111(1):43–47, 2025
2025
-
[11]
Gao and L
P. Gao and L. Zhao. Shifted moments of quadratic DirichletL-functions.Canad. Math. Bull., 68(4):1116– 1143, 2025
2025
-
[12]
Gao and L
P. Gao and L. Zhao. Bounds for moments of twisted Fourier coefficients of modular forms. Preprint. arXiv:2412.12515
-
[13]
Gao and Y
P. Gao and Y. Zhao. Mean squares of quadratic twists of the Fourier coefficients of modular forms.The Ramanujan Journal, 70(3):42, 2026
2026
-
[15]
Goldfeld and X
D. Goldfeld and X. Li. A standard zero free region for rankin–selbergL-functions.International Mathe- matics Research Notices, 2018(22):7067–7136, 2018
2018
-
[16]
Goldfeld and C
D. Goldfeld and C. Viola. Mean values ofL-funtions associated to elliptic, fermat and other curves at the centre of the critical strip.Journal of Number Theory, 11(3):305–320, 1979
1979
-
[17]
A. J. Harper. Sharp conditional bounds for moments of the Riemann zeta function. Preprint. arXiv:1305.4618
-
[18]
D. R. Heath-Brown. A mean value estimate for real character sums.Acta Arith., 72(3):235–275, 1995
1995
-
[19]
Iwaniec and E
H. Iwaniec and E. Kowalski.Analytic Number Theory, volume 53 ofAmerican Mathematical Society Col- loquium Publications. American Mathematical Society, Providence, 2004
2004
-
[20]
M. Jutila. On the mean value ofL(1/2, χ) for real characters.Analysis, 1(2):149–161, 1981
1981
-
[21]
A. A. Karatsuba.Basic analytic number theory. Springer Science & Business Media, 2012
2012
-
[22]
J. P. Keating and N. C. Snaith. Random matrix theory andL-functions ats= 1/2.Comm. Math. Phys., 214(1):91–110, 2000
2000
-
[23]
Koukoulopoulos
D. Koukoulopoulos. Pretentious multiplicative functions and the prime number theorem for arithmetic progressions.Compos. Math., 149(7):1129–1149, 2013
2013
-
[24]
H. L. Montgomery and R. C. Vaughan.Multiplicative number theory. I. Classical theory, volume 97 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2007
2007
-
[26]
G. Shimura. On the holomorphy of certain Dirichlet series.Proc. London Math. Soc. (3), 31(1):79–98, 1975
1975
-
[27]
Soundararajan
K. Soundararajan. Moments of the Riemann zeta function.Ann. of Math. (2), 170(2):981–993, 2009
2009
-
[28]
B. Szab´ o. High moments of theta functions and character sums.Mathematika, 70(2):Paper No. e12242, 37 pp., 2024
2024
Reviewed August 7, 2026 · model on record in the stance chip above.
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