{"id":"d4f9838b-2074-4194-850a-8b8a8746246d","arxiv_id":"2507.18186","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"An asymptotic formula with power-saving error is established for the twisted fourth moment of Dirichlet L-functions to a fixed prime power modulus q=q0^n0 with n0 at least 50.","lead":"This paper proves an asymptotic formula for the twisted fourth moment of Dirichlet L-functions averaged over primitive characters modulo a prime power, with a power-saving error term. It extends results known for prime moduli to a new family of moduli and gives explicit main terms.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Section 5 relies on an unproved prime-power variant of [5, Thm. 5] (Lemma 2.12); if it fails for r|q, s|r, the S_±,2 bound and the power saving collapse. The Section 10 exponent line is also written incorrectly, though a sharper q1 choice repairs it.","rationale":"The reader's strongest claim is Theorem 1.1. The most insecure point in the proof is Lemma 2.12, because it is unproved and is genuinely a variation of a cited theorem, applied in the prime-power setting r|q, s|r. If Lemma 2.12 holds, Proposition 5.1 and the final power saving follow; if it fails, no adjustment elsewhere in the paper repairs Section 5. The Section 10 exponent-balance issue is real as written, but it is a minor gap: since q1 is an integral power of q0, the minimal choice i0=floor(n0/4) gives q1^{-1/4}=q^{-i0/(4n0)} <= q^{-67/1152} for n0>=50, which validates the stated exponent. I therefore disagree with the reader's claim that n0>=58 is needed. The unproved Lemma 2.12 justifies keeping the verdict conditional rather than accepting outright, but it does not justify rejection without checking [5]. Hence the reader's conditional verdict remains unchanged.","tokens_in":42648,"tokens_out":22986,"duration_ms":229675,"concrete_test":"Re-derive Lemma 2.12 from [5, Theorem 5] for prime-power moduli r=q0^j and s=q0^i with i<=j, checking hypotheses such as squarefreeness or (r,s)=1; if the derivation succeeds, Section 5's bound is supported, and if it fails, the offending step identifies exactly where the S_±,2 estimate and hence the q^{1-1/576} power saving break down.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Lemma 2.12 is the load-bearing input for the far-apart off-diagonal estimate. In Section 5, after Voronoi summation, (5.8)-(5.9) apply Lemma 2.12 to bound S'_+,M,N, and (5.10)-(5.13) lead to Proposition 5.1; Proposition 5.1 then enters the final error R in (10.2). The paper states Lemma 2.12 as a 'slight variation' of [5, Theorem 5] and gives no proof. The application requires the bound for r=q0^j, s=q0^i with i<=j, i.e. for prime-power moduli where r and s share the prime q0; this is exactly the setting not obviously covered by a theorem whose proof may use squarefree factorization, injectivity of reduction, or (r,s)=1. If Lemma 2.12 fails in this range, the S_±,2 contribution cannot be bounded by the advertised power saving, so Theorem 1.1 as stated is unsupported. This is a correctness risk. Secondary: the displayed inequality in Section 10, q^{1+η0/2+η1/2}/q1^{1/4} <= q^{1+1/1152+1/18-1/16+1/(4n0)} <= q^{1-1/576}, is false as written for n0<58; however, because q1=q0^{i0} is an integral power, choosing i0=floor(n0/4) gives q1^{-1/4} <= q^{-67/1152} for n0>=50, so the intended exponent can be recovered. Thus the reader's n0>=58 worry is not fatal; the unproved Lemma 2.12 is the real concern.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper evaluates, for a fixed prime-power modulus q = q0^{n0} with n0 >= 50, the twisted shifted fourth moment of even primitive Dirichlet L-functions, proving an asymptotic formula with a six-term main term and power-saving error O(q^{1-1/576} + q^{1-1/n0}) under the size condition (1.8). The proof combines the approximate functional equation for products of L-functions, character orthogonality, a dyadic decomposition, Voronoi summation, a Kloosterman large sieve in the far-apart case, and the delta method together with the Kuznetsov trace formula and spectral large sieve in the close-range case, following the frameworks of Blomer-Milicevic, Hough, Zacharias, and Liu. The paper is a substantial technical extension of existing fourth-moment results to fixed prime powers.","tokens_in":43014,"tokens_out":9003,"duration_ms":82871,"significance":"If the result is valid, it is a significant extension of the twisted fourth moment for Dirichlet L-functions from prime and factorable moduli to a fixed prime-power modulus, with an explicit main term matching the CFKRS conjecture and a power-saving error. The paper is technically demanding and carefully organized, and it makes clear which ingredients are imported from prior work. The main term and the broad structure of the argument are credible. However, the proof relies on an unproved large-sieve inequality for Kloosterman sums in a prime-power setting, and the final exponent balance in Section 10 is written incorrectly for the stated range n0 >= 50; both issues are load-bearing for Theorem 1.1 and need to be addressed before the claim can be accepted.","major_comments":[{"comment":"Lemma 2.12 is the critical input for the far-apart off-diagonal estimate. It is applied in (5.8)-(5.9) with r | d | q and s = (r, q1), and it leads directly to Proposition 5.1 and then to the term q^{1+eta0/2+eta1/2} q1^{-1/4} in the final error R in (10.2). The lemma is stated as a 'slight variation' of [5, Theorem 5] and no proof is given. The application requires r and s to be powers of the same prime q0 (since r | q0^{n0} and s | r), so r and s are not coprime and the modulus is not squarefree; this is precisely a range where a variation of a theorem proved in a different setting needs independent verification. If Lemma 2.12 fails in this range, the bound for S_{+,2} + S_{-,2} collapses and Theorem 1.1 is unsupported. The authors should provide a complete proof of Lemma 2.12 or a precise reference that covers prime-power levels with s | r and (r/s, 2) = 1.","section":"Lemma 2.12 / Section 5, (5.8)-(5.10), Proposition 5.1"},{"comment":"The displayed chain q^{1+eta0/2+eta1/2}/q1^{1/4} <= q^{1+1/1152+1/18-1/16+1/(4n0)} <= q^{1-1/576} is not correct as written for n0 = 50. With eta0 = 1/576 and eta1 = 1/9, the second inequality requires 1/1152 + 1/18 - 1/16 + 1/(4n0) <= -1/576, which holds only for n0 >= 58. The conclusion can be repaired, however, by choosing i0 = floor(n0/4) and using q1^{-1/4} = q^{-i0/(4n0)}; for n0 >= 50 this gives i0/(4n0) >= 67/1152, so the desired exponent q^{1-1/576} follows. The authors should either correct the displayed inequality with the sharper choice of q1 or adjust the statement of Theorem 1.1 to the range n0 >= 58.","section":"Section 10, exponent balance after (10.2)"}],"minor_comments":[{"comment":"There are several typographical errors that should be corrected: 'focuse' in Section 1, 'Eisentein' in Section 2.9, 'simliar' in Section 3, 'supscript' in Section 5, and 'Tthe' at the beginning of Section 7.","section":"Section 1, Section 2.9, Section 3, Section 5, Section 7"},{"comment":"The phrase 'a slight variation of given [5, Theorem 5]' is grammatically incomplete; it should read 'a slight variation of [5, Theorem 5]'.","section":"Lemma 2.12"},{"comment":"The notation 'M,N ≪ log q' in (3.17) is shorthand for dyadic ranges with O(log q) choices; this should be stated explicitly to avoid ambiguity.","section":"Section 3, (3.17)"},{"comment":"The notation 'T ±;∗∗∗∗ ±,M,N' is introduced without an explicit definition of the four sign patterns; a sentence explaining that ∗∗∗∗ ranges over the four combinations appearing in (6.6) would improve readability.","section":"Section 9, (9.1)"}],"recommendation":"major_revision","confidential_remarks":"The unproved Lemma 2.12 is the main correctness risk. I would ask for a complete proof of that large-sieve inequality in the prime-power setting before considering the paper for publication; the Section 10 exponent issue, by contrast, is a repairable local error. The paper's reliance on [5, Theorem 5] should be made explicit in the introduction and the proof, since the advertised n0 >= 50 range depends on it."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The headline: this is a new result, the first asymptotic evaluation of the twisted fourth moment of Dirichlet L-functions for fixed prime-power moduli, with a power-saving error term. The paper does a solid job of assembling machinery from Blomer–Milićević, Liu, and Zacharias, with explicit main terms. Diagonal, close-off-diagonal, and far-off-diagonal treatments follow familiar templates; the use of the Kuznetsov formula and arithmetic large sieve is standard. If correct, it gives a useful tool for moment bounds below the fourth and for mollified moments.\n\nThe genuine soft spot is Lemma 2.12. It is stated as a 'slight variation' of [5, Theorem 5] and it carries the far-apart off-diagonal estimate in Section 5. The application needs the bound for r|q, s|r with r and s sharing the prime q0, exactly the setting where a proof written for squarefree or coprime moduli might break. The authors give no proof or even a sketch. That is a real gap, not a formality; if the lemma fails in this range, the advertised power saving collapses. I would want the referee to verify this variation or obtain a complete proof before the theorem is taken as stated.\n\nThe Section 10 exponent balance is a minor issue. The displayed inequality as written appears to require n0 at least 58 for the stated q^{1-1/576} error, not n0 at least 50. But the stress-test is right that this is repairable: with q1 an exact power of q0, taking i0=floor(n0/4) makes q1^{-1/4} small enough for n0 at least 50, so the intended exponent is recovered. I would not block on this.\n\nThe citation pattern is fine; the proof leans on external results, not on a chain of the authors' own papers. The paper is clearly written by people who know the literature.\n\nMy recommendation: send it to peer review. The theorem is new and the machinery is appropriate; the unproved Lemma 2.12 is the kind of thing a good referee can resolve. I would want to see that resolved before citing the result as stated.","headline":"New result for prime-power moduli, but the far-apart off-diagonal bound rests on an unproved variation of a cited large sieve; exponent typo is minor.","tokens_in":43581,"tokens_out":5891,"would_cite":true,"duration_ms":51142,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M06"],"pacs":[],"model":"deepseek-v4-flash","headline":"For prime-power moduli with exponent at least 50, the twisted fourth moment of Dirichlet $L$-functions is evaluated asymptotically with a power-saving error term.","keywords":["twisted fourth moment","Dirichlet L-functions","prime power modulus","shifted moments","approximate functional equation","Kloosterman sums","Kuznetsov trace formula","power-saving error term"],"falsifier":"Check Lemma 2.12 directly for $q=q_0^{n_0}$ with $n_0$ between 50 and 57: if the claimed bound fails for some $r\\mid q$, $s\\mid r$, then the error term in Theorem 1.1 is not obtained. Also evaluate the final exponent inequality in Section 10 with $n_0=50$ to see whether $q^{1+1/1152+1/18-1/16+1/(4n_0)} \\le q^{1-1/576}$ actually holds; if not, the stated range $n_0\\ge 50$ is false as written.","tokens_in":42393,"feed_emoji":"🧮","tokens_out":12599,"duration_ms":110796,"temperature":0.7,"pith_summary":"This paper proves an asymptotic formula for the twisted fourth moment of Dirichlet $L$-functions at the central point, for moduli that are powers of a single odd prime. For $q=q_0^{n_0}$ with $n_0\\ge 50$, the sum over primitive even characters of four shifted $L$-values times $\\chi(a)\\chi(b)$ is shown to equal six explicit main terms plus an error of size $q^{\\varepsilon}(1+|\\alpha|)^4(1+|\\beta|)^4(1+|\\gamma|)^4(1+|\\delta|)^4(ab)^7(q^{1-1/576}+q^{1-1/n_0})$. The main terms match the shape of the conjectured fourth-moment formula, and the error term is a genuine power saving over the trivial size. The result extends earlier prime-modulus evaluations to prime powers and supplies the kind of uniform shifted-moment asymptotics needed for mollification and upper-bound work.","feed_headline":"Prime-power fourth moment evaluated with power saving","feed_subtitle":"Prime-power moduli are now covered with an explicit error term and six main terms.","key_machinery":"The central machinery is the approximate functional equation for a product of four Dirichlet $L$-functions (Lemma 2.5), which converts the moment into two double sums over $m,n$ with smooth weights. Character orthogonality (Lemma 2.2) turns the character sum into congruence conditions $ma\\pm nb\\equiv 0 \\pmod d$. The off-diagonal sums are then split: when the summation lengths $M$ and $N$ are far apart, Voronoi summation and a large sieve inequality for Kloosterman sums (Lemma 2.12) control them; when $M$ and $N$ are close, the $\\delta$-method detects the congruence, and a second Voronoi summation followed by the Kuznetsov trace formula, which relates sums of Kloosterman sums to spectral data of automorphic forms, and a spectral large sieve bounds the remainder. The final power saving comes from optimizing two parameters $\\eta_0=1/576$ and $\\eta_1=1/9$.","core_discovery":"The discovery is Theorem 1.1: under the size condition $(1+|\\alpha|)^4(1+|\\beta|)^4(1+|\\gamma|)^4(1+|\\delta|)^4(ab)^7 \\ll q^{\\min(1/576,1/n_0)-\\varepsilon_0}$, the twisted fourth moment $S(\\alpha,\\beta,\\gamma,\\delta;a,b)$ equals the sum of the six terms $S_1,\\dots,S_6$ in (1.10) plus a power-saving error. Each of the six terms is an explicit product of zeta factors, powers of $a$ and $b$, and the multiplicative coefficients $\\tau_{\\alpha,\\beta,\\gamma,\\delta}$, reflecting a distinct pairing of the four shifts; their total is the natural continuation of the known prime-modulus formula and agrees with the conjectured integral-moment formula. The proof obtains this by writing the four-$L$-function product through an approximate functional equation, detecting character orthogonality by congruences, and splitting the resulting sums into diagonal, far-apart, and close-proximity regimes.","pith_inferences":["In the editor's reading, the unproved large-sieve inequality for Kloosterman sums is the main obstacle to lowering the exponent threshold; a proof for all $r\\mid q$, $s\\mid r$ would likely push $n_0$ well below 50.","The uniform polynomial dependence on $ab$ suggests the formula is ready to be used as an input for mollified fourth moments with short mollifiers; for twists of size comparable to $q$, a different treatment would be needed.","One testable extension is to check the six-term main shape numerically for small prime powers and small shifts, which would separate the analytic error-term mechanism from any hidden issue in the quoted large sieve."],"forward_implications":["For every odd prime $q_0$ and every exponent $n_0\\ge 50$, the twisted fourth moment has an asymptotic formula with error $q^{1-1/576}+q^{1-1/n_0}$, a genuine power saving over the main term.","The six explicit main terms reproduce the structure of the conjectured moment formula, so the result gives a concrete check of that conjecture for prime-power families.","The uniform polynomial dependence on the shifts and on $ab$ makes the formula usable as an input for mollified moments and for upper bounds below the fourth moment.","The proof splits the off-diagonal contribution by the relative size of $m$ and $n$, combining elementary congruences, Voronoi summation, and spectral theory in a way that can serve as a template for other families with fixed prime-power conductor."],"supporting_citations":[{"why":"Supplies the large sieve inequality for Kloosterman sums quoted as Lemma 2.12 and the strategy for the far-apart regime.","marker":"[5]"},{"why":"Provides the approximate functional equation for products of L-functions used in Lemma 2.5 and the prime-modulus fourth moment framework.","marker":"[24]"},{"why":"Provides the Voronoi summation formula for divisor sums, the delta-method setup, and the computation of the secondary main term.","marker":"[15]"},{"why":"Provides the Kuznetsov trace formula, spectral large sieve inequalities, and the treatment of the close-proximity error terms.","marker":"[25]"},{"why":"Provides the delta-method identity used to detect the congruence condition in the off-diagonal sums.","marker":"[8]"},{"why":"Provides the orthogonality relation for even Dirichlet characters used as Lemma 2.2.","marker":"[21]"},{"why":"States the conjectured integral-moment formula whose main-term structure the six terms reproduce.","marker":"[7]"}],"fun_headline_variants":["Twisted fourth moment for prime power moduli","Explicit six-term formula with power saving error","Prime-power Dirichlet L-function moment solved","Fourth moment asymptotics for prime power modulus"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof rests on a large-sieve inequality for Kloosterman sums modulo prime powers (Lemma 2.12) that is quoted as a variation of a known result rather than proved, and separately the final exponent balance appears to need $n_0 \\ge 58$ rather than the stated $n_0 \\ge 50$.","fun_headline_variants_meta":{"raw":{"variants":["Twisted fourth moment for prime power moduli","Explicit six-term formula with power saving error","Prime-power Dirichlet L-function moment solved","Fourth moment asymptotics for prime power modulus"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000536,"raw_usage":{"total_tokens":2488,"prompt_tokens":774,"completion_tokens":1714,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":390,"completion_tokens_details":{"reasoning_tokens":1657}},"tokens_in":390,"tokens_out":1714,"duration_ms":14949,"temperature":1.0,"reasoning_tokens":1657,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T14:39:48.485250+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Check Lemma 2.12 directly for $q=q_0^{n_0}$ with $n_0$ between 50 and 57: if the claimed bound fails for some $r\\mid q$, $s\\mid r$, then the error term in Theorem 1.1 is not obtained. Also evaluate the final exponent inequality in Section 10 with $n_0=50$ to see whether $q^{1+1/1152+1/18-1/16+1/(4n_0)} \\le q^{1-1/576}$ actually holds; if not, the stated range $n_0\\ge 50$ is false as written.","supporting_citations":[{"cited_title":"Blomer and D","cited_arxiv_id":null,"evidence_quote":"Supplies the large sieve inequality for Kloosterman sums quoted as Lemma 2.12 and the strategy for the far-apart regime."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the approximate functional equation for products of L-functions used in Lemma 2.5 and the prime-modulus fourth moment framework."},{"cited_title":"Liu, Zeros and moments of L-functions and applications , 2024","cited_arxiv_id":null,"evidence_quote":"Provides the Voronoi summation formula for divisor sums, the delta-method setup, and the computation of the secondary main term."},{"cited_title":"Zacharias, Mollification of the fourth moment of Dirichlet L-functions, Acta Arith","cited_arxiv_id":null,"evidence_quote":"Provides the Kuznetsov trace formula, spectral large sieve inequalities, and the treatment of the close-proximity error terms."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the delta-method identity used to detect the congruence condition in the off-diagonal sums."},{"cited_title":"Soundararajan, The fourth moment of Dirichlet L-functions, in: Analytic number theory, 239–246, Clay Math","cited_arxiv_id":null,"evidence_quote":"Provides the orthogonality relation for even Dirichlet characters used as Lemma 2.2."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"States the conjectured integral-moment formula whose main-term structure the six terms reproduce."}],"review_version":1}