{"id":"67a353ce-d0c1-4558-9f68-21e9a27f1ebe","arxiv_id":"2412.12515","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Under GRH, moments of character-twisted sums of Fourier coefficients of a fixed modular form are bounded by the number of characters times Y^m times a logarithmic power, matching the character-sum bounds.","lead":"This paper proves upper bounds, assuming the generalized Riemann hypothesis, for averages over Dirichlet characters of powers of sums of modular form Fourier coefficients. The bounds match the size previously known for plain character sums and give a conditional estimate for how modular form coefficients distribute over character twists.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.5 rests on estimate (4.12), asserted without proof; the adaptation from [12, Prop. 3] and the dyadic summation that must yield exponent (m−1)^2 are not demonstrated.","rationale":"The paper is a plausible extension of the Soundararajan–Harper machinery to modular-form Fourier coefficients; the shifted-moment theorems and the subsequent character-sum deductions have the right overall shape, and I found no circularity or internal contradiction in the main construction. The genuinely load-bearing weakness is the unproved integral-moment estimate (4.12) and the dyadic step that converts it into Lemma 4.2. The reader identified (4.12) as the weakest assumption; I agree, and I add that the stated justification via boundedness of g2 is incomplete and that a naive dyadic summation produces a possible extra log q. Since this is a gap in proof detail rather than a demonstrated falsity, and because the cited method may well supply the missing argument, the appropriate verdict remains CONDITIONAL rather than REJECT or ACCEPT.","tokens_in":29756,"tokens_out":17510,"duration_ms":159877,"concrete_test":"Independently re-derive (4.12) from Corollary 1.3, following the proof of [12, Proposition 3] line by line for L(s,f⊗χ). Then explicitly perform the dyadic summation of Section 4.4 with B=e^n and weight (1+|t|)^{-1}, checking whether the total logarithmic exponent is (m−1)^2 or (m−1)^2+1. If the latter, Theorem 1.5's stated power is false; if the former, the missing summation step should be written out.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Lemma 4.2 depends entirely on (4.12): the bound on ∑_χ (∫_0^B |L(1/2+it,f⊗χ)|dt)^{2m}. The paper says this follows by a 'straightforward modification' of [12, Proposition 3] using Corollary 1.3 with g2 ≤ (log log B)^{O(1)}. This is not a routine specialization. First, Corollary 1.3 is stated for a fixed integer k and exponents a_j, while (4.12) needs real m>2; the interpolation is not shown. Second, the asserted bound on g2 is only valid when B<q, whereas (4.12) is stated for B=q^{O(1)}; for x≥eq, (1.4) gives g2(x)=log q, so the stated justification does not cover its own hypothesis. Third, even in the intended range B≤q^ε, the B^3 and B^{2m} factors in (4.12) are not automatically harmless. After the standard dyadic splitting of ∫_{|t|≤q^ε}|L|/(1+|t|)dt, the first term of (4.12) produces a convergent sum ∑e^{(3−2m)n}, but the second term produces roughly ∑_{n≤ε log q} 1, an extra log q, unless additional saving is supplied. The reference to [12, Section 3] may contain that saving, but it is not reproduced in the modular-twist setting. A further sign of the gap is that (4.13) is dimensionally inconsistent: its left side is independent of Y while the right side contains Y^m. Thus the central bound Theorem 1.5 is conditional on an unverified analytic estimate.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":30089,"tokens_out":50338,"duration_ms":397009,"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":[{"comment":"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.","section":"§4.4, Eqs. (4.12)–(4.13)"},{"comment":"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.","section":"§3, around Eq. (3.6); Lemma 2.8"},{"comment":"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.","section":"§3, Eq. (3.7)"},{"comment":"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.","section":"§3, Eqs. (3.1)–(3.3)"}],"minor_comments":[{"comment":"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.","section":"§4.4, Eq. (4.12)"},{"comment":"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.","section":"§5.3, Eq. (5.5)"},{"comment":"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.","section":"§2.1, Eq. (2.6)"},{"comment":"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.","section":"§2.2, Eq. (2.34)"},{"comment":"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.","section":"§4.5, Lemma 4.3"},{"comment":"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.","section":"§2.9, Lemmas 2.15–2.16"},{"comment":"The second displayed line of (3.7) contains a duplicated summation symbol '∑_{p≤X}∑_{p≤X}'; one of the two should be removed.","section":"§3, Eq. (3.7)"}],"recommendation":"major_revision","confidential_remarks":"The manuscript delegates its two most consequential estimates to other papers: (4.12) to Szabó [12, Proposition 3 and Section 3], and the proof structure of Theorem 1.2 to the authors' own unpublished preprint [4] (arXiv:2406.18024). If [4] is still in refereeing, the editors may want to confirm that the 'straightforward modifications' used here are indeed those appearing in [4], since Theorem 1.2 of the present paper is not a formal consequence of [4, Theorem 1.1] but requires redoing the argument with modular L-functions and the A(d) weights. I would also note for the record that the dimensional inconsistency in (4.13) and the sign error in (3.7), while likely fixable, mean that Sections 3 and 4 need careful hands-on checking and not merely proofreading."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThis is a direct extension of the Soundararajan–Harper machinery to twisted modular-form coefficients. The new material is Theorems 1.5 and 1.6: GRH-conditional bounds on moments of sums of λ_f(n)χ(n) that match the logarithmic shape of Szabó’s unweighted character-sum bounds. The shifted-moment results Theorems 1.1 and 1.2 are also new and are the natural modular analogues. The paper is honest about its assumptions, and Theorem 1.6’s proof, via Proposition 5.4, is actually carried out in detail. The algebra in (3.6) and the use of (3.5) check out.\n\nThe soft spot is Lemma 4.2. Estimate (4.12) is load-bearing for Theorem 1.5 and is asserted without proof as a “straightforward modification” of [12, Proposition 3]. I don’t think that is routine. Corollary 1.3 is stated for fixed integer k and exponents a_j, while (4.12) needs real m>2; the interpolation is not shown. The claim that g2 is bounded by (log log B)^{O(1)} only holds when B<q (or B≤q^ε), but (4.12) is stated for B=q^{O(1)}; for x≥eq, g2(x)=log q. Even in the intended range, after dyadic splitting the B^{2m} term seems to produce an extra log q unless extra saving is supplied. The reference to [12, Section 3] may contain that saving, but it is not reproduced here. Also, (4.13) is dimensionally inconsistent: the left side has no Y, the right side has Y^m. That looks like a typo, but it reinforces the need for a careful rewrite of Lemma 4.2.\n\nNone of this is fatal, and I see no circularity. These are fillable gaps and typos, not a wrong central idea. A referee should ask for a full proof of (4.12) before accepting Theorem 1.5.\n\nThis paper is for analytic number theorists working on moments of L-functions and character sums. It deserves a serious referee; with Lemma 4.2 fixed it would be a useful contribution. I would send it out.","headline":"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.","tokens_in":30665,"tokens_out":6187,"would_cite":false,"duration_ms":52029,"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, twisted modular coefficient sums obey character-sum moment bounds.","keywords":["Dirichlet characters","modular L-functions","shifted moments","upper bounds","Fourier coefficients","Hecke eigenform","quadratic twists","generalized Riemann hypothesis"],"falsifier":"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.","tokens_in":29518,"feed_emoji":"🔢","tokens_out":11054,"duration_ms":89627,"temperature":0.7,"pith_summary":"This paper proves, under the generalized Riemann hypothesis (GRH), upper bounds for the moments of sums of Fourier coefficients of a fixed holomorphic Hecke eigenform twisted by Dirichlet characters. For twists with a fixed modulus $q$, the $2m$-th moment $S_m(q,Y;f)$ is shown to be at most $\\phi(q)Y^m(\\log q)^{(m-1)^2}$ for $m>2$ and $Y\\le q$; for quadratic twists, $T_m(X,Y;f)$ is at most $XY^m(\\log X)^{2m^2-3m+2}$ for $m\\ge2$ and $Y\\le X$. These match the leading shape of the sharp high-moment bounds for ordinary character sums, with the diagonal factor $Y^m$ times a power of a logarithm. The proof route is to establish shifted-moment estimates for the twisted modular $L$-functions and then convert them into bounds on the coefficient sums by smoothing, Mellin inversion, and dyadic decomposition.","feed_headline":"Moment bounds for twisted modular coefficients match character sums","feed_subtitle":"Under GRH, character-twisted Fourier coefficient sums stay within log powers of the diagonal size.","key_machinery":"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.","core_discovery":"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 $L(s,\\mathrm{sym}^2 f)$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the high-moment character-sum bound that the paper generalizes, and the integral-moment estimate that (4.12) is said to modify.","marker":"[12]"},{"why":"Establishes shifted moments for quadratic Dirichlet $L$-functions, the template for Theorem 1.2 and Lemma 2.8.","marker":"[4]"},{"why":"Provides the refined dyadic decomposition used to turn pointwise log bounds into moment majorants.","marker":"[5]"},{"why":"Originates the resonance method for conditional moment bounds on which the shifted-moment estimates rest.","marker":"[11]"},{"why":"Gives fixed-modulus modular $L$-function moment bounds and the mean value of $\\lambda_f(p)^2$ used in Lemma 2.4.","marker":"[3]"},{"why":"Supplies the large-sieve estimate for real character sums used to control the unsmoothed tail in Lemma 5.2.","marker":"[6]"},{"why":"Gives $|\\alpha_p|=|\\beta_p|=1$ via the Weil-conjecture proof, controlling prime-power coefficients throughout.","marker":"[2]"},{"why":"Provides holomorphy of the symmetric-square $L$-function, used in the prime sum estimates.","marker":"[10]"}],"fun_headline_variants":["Twisted modular moments: GRH pins log-power bounds","Log-power moment bounds for twisted modular sums under GRH","GRH implies sharp log exponents for twisted Fourier sums","Modular L-function shifts yield character-sum moment bounds","Under GRH, twisted modular moments stay within log powers"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Twisted modular moments: GRH pins log-power bounds","Log-power moment bounds for twisted modular sums under GRH","GRH implies sharp log exponents for twisted Fourier sums","Modular L-function shifts yield character-sum moment bounds","Under GRH, twisted modular moments stay within log powers"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000692,"raw_usage":{"total_tokens":3054,"prompt_tokens":792,"completion_tokens":2262,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":408,"completion_tokens_details":{"reasoning_tokens":2183}},"tokens_in":408,"tokens_out":2262,"duration_ms":14874,"temperature":1.0,"reasoning_tokens":2183,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T14:00:21.883798+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Soundararajan, Moments of the Riemann zeta function , Ann","cited_arxiv_id":null,"evidence_quote":"Originates the resonance method for conditional moment bounds on which the shifted-moment estimates rest."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives fixed-modulus modular $L$-function moment bounds and the mean value of $\\lambda_f(p)^2$ used in Lemma 2.4."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the large-sieve estimate for real character sums used to control the unsmoothed tail in Lemma 5.2."}],"review_version":1}