{"id":"a02d5a71-2bb3-4809-ac74-58a5fe4f1a5e","arxiv_id":"2506.22256","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"An unconditional asymptotic formula is obtained for the smoothed mean square of quadratic twists of Fourier coefficients of a modular form.","lead":"This paper derives an asymptotic formula for a smoothed average of the squares of character sums whose terms are weighted by Fourier coefficients of a modular form. The result is an unconditional extension of earlier work that handled the unweighted case.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The main term rests on identifying the paper's Euler product Z(u,v) with the quoted factorization from [14,(4.6)]; this identification is not proved and should be verified locally.","rationale":"The paper is a plausible adaptation of the Soundararajan-Young method, and the claimed error terms balance correctly when Z = sqrt(X/Y). The most structurally critical step is the identification of the series Z(u,v) with the factorization quoted from [14,(4.6)]; the reader's weakest-assumption analysis correctly identifies this. I did not find an internal contradiction in the contour shifts themselves: the pole at v = 1 - u is crossed exactly once, the later shift of u does not cross a second pole for small epsilon, and the convexity estimates appear sufficient for the stated error XY^{1/2+epsilon}. A secondary issue, also noted by the reader, is that Theorem 1.1 says C0 depends only on Phi and Psi, while (3.15) visibly contains L(2u,sym² f), L(2-2u,sym² f), L(1,sym² f), and Z2, all of which depend on f; this is a statement error in the theorem, not a flaw in the asymptotic for a fixed f. Because the central concern is the unverified Euler-product identification, and a direct local check can settle it, the appropriate verdict remains conditional acceptance as the reader recommended.","tokens_in":9967,"tokens_out":27379,"duration_ms":283596,"concrete_test":"For a prime p and a pair (α,β) with αβ = 1 and α + β = λ_f(p), compute the p-local factor Z_p(u,v) = p/(p+1) * (1/2)[ A(p^{-u})A(p^{-v}) + A(-p^{-u})A(-p^{-v}) ], where A(t) = ((1-αt)(1-βt))^{-1}. Compare it with the local factor of ζ(u+v)L(2u,sym² f)L(2v,sym² f)L(u+v,sym² f), and show the quotient Z2,p(u,v) is 1 + O(p^{-1/2-δ}) uniformly for Re u, Re v > 1/4 + ε. Repeat for several primes, e.g., p = 3, 5, 7, and at least one non-real λ (e.g., λ = 0). If any p fails, the quoted [14,(4.6)] factorization does not apply to this Z and the main term collapse.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1.1 is proved by shifting contours in (3.13), and the entire main term is the residue at v = 1 - u of Z(u,v). The series Z(u,v) defined after (3.13) has an explicit extra Euler factor ∏_{p|n1n2} p/(p+1) and the square-product condition n1n2 = □. The paper says this is 'essentially' the function on [14, p. 1108] with u,v shifted by 1/2, and then imports the factorization [14,(4.6)]: Z(u,v) = ζ(u+v)L(2u,sym² f)L(2v,sym² f)L(u+v,sym² f)Z2(u,v), with Z2 uniformly bounded for Re u, Re v > 1/4. This quoted identity is load-bearing: it fixes the pole that produces the XY main term and also determines the constant C0 in (3.15). If the p/(p+1) factor or the square condition changes the local Euler factors relative to [14], the factorization may fail, or the stated region of uniform boundedness may be smaller; then the contour shifts in Section 3.2 would not yield the claimed asymptotic. The paper does not reproduce the Euler-product computation, only says 'essentially'.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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).","tokens_in":10199,"tokens_out":52198,"duration_ms":530574,"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":[{"comment":"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.","section":"Section 3.2, after (3.13)"},{"comment":"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.","section":"Equation (3.11)"},{"comment":"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.","section":"Theorem 1.1 and equation (3.15)"}],"minor_comments":[{"comment":"The phrase 'compacted supported functions' should read 'compactly supported functions'.","section":"Introduction, page 1"},{"comment":"There is a duplicated article in 'the the O-term'; this should be corrected.","section":"Equation (3.7)"},{"comment":"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.","section":"Notation throughout Section 3"},{"comment":"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).","section":"Theorem 1.1"},{"comment":"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.","section":"Equation (3.20)"}],"recommendation":"major_revision","confidential_remarks":"The paper is a straightforward application of the Soundararajan–Young template, and the central claim is likely correct. The main issues are the unproved identification with [14, (4.6)] and a likely typo in the normalization of \\tilde H_0 in (3.11). Both are fixable within the manuscript's scope, but they are load-bearing for the main term, so I cannot recommend acceptance without seeing the corrected and verified versions."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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].\n\nThere 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.\n\nThe 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.\n\nOverall, 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.","headline":"A credible, useful extension of the Soundararajan-Young method that needs one Euler-product verification before the main term is fully supported; worth refereeing.","tokens_in":10732,"tokens_out":3693,"would_cite":true,"duration_ms":37114,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11N37","11L05","11L40"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["mean square","quadratic Dirichlet character","modular L-functions","quadratic twists","Fourier coefficients of cusp forms","symmetric square L-function","Poisson summation","asymptotic formula"],"falsifier":"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).","tokens_in":9730,"feed_emoji":"","tokens_out":9777,"duration_ms":107020,"temperature":0.7,"pith_summary":"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.","feed_headline":"Twisted modular coefficients get an unconditional mean-square law","feed_subtitle":"The leading term is a constant times XY, with explicit power-saving error; no unproved hypothesis is needed.","key_machinery":"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].","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the factorization of $Z(u,v)$ and the second-moment bound for quadratic twists of modular $L$-functions on which both the main term and the off-diagonal estimates rely.","marker":"[14]"},{"why":"Supplies the Poisson summation formula and the evaluation of Gauss sums used to transform the sum over $d$.","marker":"[12]"},{"why":"Gives the smoothed second-moment framework and the earlier asymptotic for real character sums that this paper extends to $\\lambda_f$-weighted sums.","marker":"[3]"},{"why":"Provides the analytic facts and convexity bounds for $\\zeta(s)$ and $L(s,\\mathrm{sym}^2 f)$ used in the contour shifts.","marker":"[8]"},{"why":"Establishes holomorphy of the symmetric square $L$-function, needed for the residue computation that produces the main term.","marker":"[11]"},{"why":"Provides the divisor-function bound $d(n)\\ll n^\\varepsilon$ used to estimate the error term in the $k=0$ contribution.","marker":"[9]"}],"fun_headline_variants":["Unconditional mean-square law for twisted modular coefficients","Quadratic twists of modular forms: exact mean-square growth","Mean-square of twisted coefficients: explicit leading term","Twisted modular sums: power-saving error, no hypotheses","New asymptotic for twisted moments of Fourier coefficients"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Unconditional mean-square law for twisted modular coefficients","Quadratic twists of modular forms: exact mean-square growth","Mean-square of twisted coefficients: explicit leading term","Twisted modular sums: power-saving error, no hypotheses","New asymptotic for twisted moments of Fourier coefficients"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00048,"raw_usage":{"total_tokens":2357,"prompt_tokens":906,"completion_tokens":1451,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":522,"completion_tokens_details":{"reasoning_tokens":1377}},"tokens_in":522,"tokens_out":1451,"duration_ms":9662,"temperature":1.0,"reasoning_tokens":1377,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T22:07:33.813344+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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).","supporting_citations":[{"cited_title":"Soundararajan and M","cited_arxiv_id":null,"evidence_quote":"Supplies the factorization of $Z(u,v)$ and the second-moment bound for quadratic twists of modular $L$-functions on which both the main term and the off-diagonal estimates rely."},{"cited_title":"Soundararajan, Nonvanishing of quadratic Dirichlet L-functions at s = 1 2 , Ann","cited_arxiv_id":null,"evidence_quote":"Supplies the Poisson summation formula and the evaluation of Gauss sums used to transform the sum over $d$."},{"cited_title":"The mean square of real character sums","cited_arxiv_id":"1908.07922","evidence_quote":"Gives the smoothed second-moment framework and the earlier asymptotic for real character sums that this paper extends to $\\lambda_f$-weighted sums."},{"cited_title":"Shimura, On the holomorphy of certain Dirichlet series , Proc","cited_arxiv_id":null,"evidence_quote":"Establishes holomorphy of the symmetric square $L$-function, needed for the residue computation that produces the main term."}],"review_version":1}