{"id":"716b93d5-a453-4d50-97ea-a8079b67f084","arxiv_id":"2608.05961","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"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.","lead":"The paper bounds, under the Generalized Riemann Hypothesis, the size of sums of modular form coefficients twisted by quadratic characters of prime modulus. It obtains the exact logarithmic growth order for smoothed even integer moments, which serves as a benchmark for random-matrix predictions in this number-theoretic family.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.2's sharp upper bound depends wholly on the imported estimate (5.3) from the authors' preprint [14], which is neither proved nor sketched here; if that estimate carries an extra log factor, the claimed exponent fails.","rationale":"The reader's verdict of CONDITIONAL is the right one. The paper has real independent content: Theorem 2.1 is proved in Section 3, and the lower-bound strategy in Section 6 uses Lemma 2.6 and Lemma 2.4 in a plausible way. However, the decisive estimate for the smoothed upper bound is imported verbatim from the authors' own unpublished preprint [14]. The passage at (5.2)-(5.3) is explicit that the bound is not proved here. Since the central claim in Theorem 1.2 collapses if (5.3) is wrong, the concern is load-bearing and is exactly what the reader identified. I also note the abstract's unsmoothed-order-of-magnitude statement is not supported by the stated theorems, but that is an overclaim about the results rather than a flaw in the main proof. Keeping the verdict CONDITIONAL with this explicit check is appropriate.","tokens_in":23327,"tokens_out":4205,"duration_ms":39472,"concrete_test":"Independently re-derive (5.3), starting with m = 4 and k = 3, directly from the definition of I_{m,B,k} in (5.2) and the explicit function g1 in (2.20), without citing [14, Theorem 1.1]; check whether the resulting exponent is exactly m(m-3)/2 = 2 and whether the B-dependence is exactly B^2. If the calculation yields (log X)^{2+c} or B^{2+c} for any c > 0, then Theorem 1.2's sharp bound is not established by this paper.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Agreeing with the reader's weakest_assumption: in Section 5, the proof of Theorem 1.2 reduces U_m(X,Y;f,W) to the bound (5.1) for Z_m(B,X), and then via (5.2) to the estimate (5.3), I_{m,B,k} << B^2 (log X)^{m(m-3)/2} (log log B)^{O_m(1)}. The text says only that this was shown in [14] during the proof of [14, Theorem 1.1]; no proof or sketch of (5.3) appears here, and [14] is the authors' own preprint. If (5.3) is off by even one power of log X, the sharp exponent in Theorem 1.2 does not follow from the present argument, and Theorem 1.3 would then be a lower bound for a different smoothed object, not a matching upper bound. There is also a gap between the abstract and the theorems: the abstract claims the correct order of magnitude for the unsmoothed m-th moment for all real m >= 4, but Theorem 1.1 only gives an upper bound with a larger logarithmic exponent, and Theorem 1.3 is a lower bound only for smoothed, even integer m. The unsmoothed lower bound is not proved. The missing proof of (5.3) is the load-bearing issue; the abstract overstatement is secondary but should be corrected.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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].","tokens_in":23722,"tokens_out":3756,"duration_ms":34949,"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":[{"comment":"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.","section":"Section 5, Eq. (5.3)"},{"comment":"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":"Abstract vs. Theorems 1.1–1.3"},{"comment":"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.","section":"Section 2, Lemma 2.6 and Section 3, Lemma 3.1"}],"minor_comments":[{"comment":"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":"Various"},{"comment":"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.","section":"Section 4.1, Eq. (4.24)"},{"comment":"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":"References"},{"comment":"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.","section":"Section 5, Eq. (5.2)"}],"recommendation":"major_revision","confidential_remarks":"The paper's main new upper bound (Theorem 1.2) is essentially a transfer of the authors' own preprint [14] to the prime-modulus family, with the decisive estimate (5.3) cited rather than proved. This is a correctness-risk concern, not a novelty concern per se, but the editor should weigh whether the paper is sufficiently self-contained for a journal article in its current form. The detailed proof of Theorem 1.1 and Theorem 2.1 is a genuine contribution and would make a solid paper if the missing estimates are supplied or clearly delimited. I also note that the abstract overstates the results; this should be corrected irrespective of the other revisions."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a competent piece of analytic number theory that does what the title says, but you should know two things before deciding how to use it. The sharp exponent in Theorem 1.2 comes almost entirely from estimate (5.3), which is imported from the authors' preprint [14] with no proof or sketch. And the abstract's claim of \"correct order of magnitude\" for the unsmoothed m-th moment for all real m >= 4 is stronger than anything proved: Theorem 1.1 is an upper bound with a larger log exponent, and the matching lower bound (Theorem 1.3) is only for smoothed, even integer m.\n\nWhat is genuinely new: the prime-modulus setting with a fixed Hecke eigenform's coefficients. The shifted moment Theorem 2.1 is proved in detail in Section 3 and looks like the main technical contribution. Proposition 2.8 and the application in Section 4 give real content. The lower bound in Theorem 1.3 is a useful counterpart. The paper follows the template of Munsch-Toma [25] and the authors' own [14]—expected adaptation, no new principle.\n\nWhere it is soft: the load-bearing estimate (5.3). The reduction of Theorem 1.2 to (5.3) is clear, but the estimate itself is cited, not derived. If it carries an extra log factor, the exponent fails. You can't verify the main theorem from this paper alone. That is a genuine self-containedness problem, and since [14] is not yet published, it is not just a formality. The authors should either include a full proof of (5.3) in this paper, or make [14] publicly verifiable and clearly cross-reference the exact location. The abstract overstatement is secondary but should be fixed. Some auxiliary lemmas (Lemma 3.1, Lemma 2.6) are referenced or stated with only a sketch; those are standard adaptations, and I would not hold them against the paper.\n\nBottom line: the paper is for specialists in moments of L-functions and character sums. It deserves a proper referee, but only after (5.3) is addressed. I'd be comfortable with a conditional accept once the gap is filled and the abstract is aligned with the theorems.","headline":"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.","tokens_in":24147,"tokens_out":2674,"would_cite":true,"duration_ms":22941,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11L40","11M06"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["shifted moments","modular L-function","Dirichlet characters","prime moduli","quadratic characters","Fourier coefficients","moments of L-functions","GRH"],"falsifier":"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.","tokens_in":23072,"feed_emoji":"🔢","tokens_out":14164,"duration_ms":112332,"temperature":0.7,"pith_summary":"This paper studies the moments of character-weighted sums $\\sum_{n\\le Y}\\chi_{8p}(n)\\lambda_f(n)$, where $\\lambda_f(n)$ are the Fourier coefficients of a fixed holomorphic Hecke eigenform, $\\chi_{8p}$ is the quadratic character of conductor $8p$, and $p$ ranges over odd primes. Assuming the Generalized Riemann Hypothesis, it proves that for every integer $m\\ge 4$ the smoothed moment $\\sum_{2<p\\le X}(\\log p)\\,|\\sum_{n\\ge 1}\\chi_{8p}(n)\\lambda_f(n)W(n/Y)|^m$ is bounded by $XY^{m/2}(\\log X)^{m(m-3)/2}$, and that for even $m\\ge 4$ a lower bound of the same order holds when $X^\\varepsilon\\ll Y\\ll X^{1-\\alpha}$. This pins down the logarithmic exponent conjectured for this family and shows that the moments are governed by the diagonal condition $n_1\\cdots n_m=\\square$. The result matters because extra powers of $\\log X$ that appeared in the previous best bounds are now shown to be artifacts of the method, at least for even integer moments.","feed_headline":"Prime-modulus twist moments hit exact log exponent","feed_subtitle":"Smoothed m-th moment has order X Y^{m/2}(log X)^{m(m-3)/2}, optimal for even m.","key_machinery":"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).","core_discovery":"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 $\\log X$.","pith_inferences":["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."],"forward_implications":["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 $X$."],"supporting_citations":[{"why":"Provides the integral estimate (5.3) that completes the proof of Theorem 1.2 and the diagonal moment bound behind Lemma 2.6.","marker":"[14]"},{"why":"Gives the smoothed-moment decomposition and lower-bound trick that Theorems 1.2 and 1.3 adapt to prime moduli.","marker":"[25]"},{"why":"Supplies the shifted-moment bounds for twisted modular $L$-functions used in Theorem 2.1 and the method for Theorem 1.1.","marker":"[12]"},{"why":"Supplies the prime-modulus summation technique and dyadic decomposition used in the proof of Theorem 2.1.","marker":"[29]"},{"why":"Supplies the high-moment bounds and dyadic integration scheme that control the shifted-moment sums in Section 3.","marker":"[28]"},{"why":"Refines the conditional-moment method to sharp bounds; Theorem 2.1 builds on this refinement.","marker":"[17]"},{"why":"Provides the GRH-conditional upper-bound method for moments whose extension underlies the shifted-moment estimates.","marker":"[27]"}],"fun_headline_variants":["Exact log power for prime twist moments","Matching bounds for quadratic twist moments under GRH","Log exponent pinned for prime-modulus twist moments","Optimal smoothed moment order for even twist degrees","GRH locks down log exponent for prime twists"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Exact log power for prime twist moments","Matching bounds for quadratic twist moments under GRH","Log exponent pinned for prime-modulus twist moments","Optimal smoothed moment order for even twist degrees","GRH locks down log exponent for prime twists"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000402,"raw_usage":{"total_tokens":2080,"prompt_tokens":912,"completion_tokens":1168,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":528,"completion_tokens_details":{"reasoning_tokens":1107}},"tokens_in":528,"tokens_out":1168,"duration_ms":10620,"temperature":1.0,"reasoning_tokens":1107,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T20:16:49.185576+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Gao and Y","cited_arxiv_id":null,"evidence_quote":"Provides the integral estimate (5.3) that completes the proof of Theorem 1.2 and the diagonal moment bound behind Lemma 2.6."},{"cited_title":"Munsch and Y","cited_arxiv_id":null,"evidence_quote":"Gives the smoothed-moment decomposition and lower-bound trick that Theorems 1.2 and 1.3 adapt to prime moduli."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the prime-modulus summation technique and dyadic decomposition used in the proof of Theorem 2.1."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the high-moment bounds and dyadic integration scheme that control the shifted-moment sums in Section 3."},{"cited_title":"Soundararajan","cited_arxiv_id":null,"evidence_quote":"Provides the GRH-conditional upper-bound method for moments whose extension underlies the shifted-moment estimates."}],"review_version":1}