{"id":"15ea92c6-63ea-4f0b-9d53-b5e84257f7ab","arxiv_id":"2606.19959","paper_version":1,"verdict":"UNVERDICTED","confidence":"UNKNOWN","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"An asymptotic with power-saving error is given for the twisted first moment of GL3 symmetric square L-functions, yielding non-vanishing results and even-moment lower bounds matching random matrix expectations.","lead":"This paper derives an asymptotic formula with power-saving error for the twisted first moment of symmetric square L-functions on GL3 in the spectral aspect. A smart generalist might read it to see concrete support for random matrix predictions on non-vanishing and moments of higher-rank L-functions.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's identification of the strong bounds and applicability of the asymmetric AFE matches the load-bearing step required for the claimed power-saving. The abstract presents these as the main ingredients without evident gaps, so the verdict remains UNVERDICTED pending full verification of those bounds.","tokens_in":1570,"tokens_out":220,"duration_ms":16537,"concrete_test":"Extract the precise statement of the bounds on the Kuznetsov integral transforms (likely in the section deriving the moment asymptotic) and check whether they deliver a positive power of the spectral parameter T in the error term after summation over the spectral family.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The abstract states the result follows from the GL3 Kuznetsov formula, an asymmetric approximate functional equation, and strong bounds on the associated integral transforms. The reader's weakest assumption correctly isolates the place where the power-saving error could fail. No internal inconsistency, circularity, or unsupported step is visible in the given description of the argument.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper establishes an asymptotic formula with a power-saving error term for the twisted first moment of symmetric square L-functions on GL_3 in the spectral aspect. This is applied to derive non-vanishing results and lower bounds of the expected order for even moments, supporting random matrix theory predictions for a unitary ensemble. The argument relies on the GL_3 Kuznetsov formula, an asymmetric approximate functional equation, and strong bounds for the integral transforms in the Kuznetsov formula.","tokens_in":1606,"tokens_out":395,"duration_ms":21167,"significance":"If the power-saving error term holds, the result would advance the spectral theory of L-functions on GL(3) by enabling applications to moments and non-vanishing that align with RMT predictions. The use of standard tools (Kuznetsov formula and approximate functional equations) with claimed strong bounds on transforms is a standard and appropriate approach in the field.","major_comments":[{"comment":"The power-saving error term in the main asymptotic formula depends critically on the strong bounds for the integral transforms arising from the GL_3 Kuznetsov formula; these bounds are stated as holding but require explicit verification of the saving achieved, as any shortfall would invalidate the claimed power saving in the error term.","section":"Main theorem / § on integral transforms"}],"minor_comments":[{"comment":"The abstract and introduction should clarify the precise range of the spectral parameter (e.g., the size of the Laplace eigenvalue) for which the asymptotic holds with the stated power saving.","section":null},{"comment":"Notation for the symmetric square L-function and the twisting character should be introduced with explicit references to standard definitions (e.g., from Gelbart-Jacquet or similar) to aid readability.","section":null}],"recommendation":"uncertain","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their careful reading of the manuscript and for identifying this important point regarding the integral transform bounds. We respond to the major comment below.","responses":[{"response":"We appreciate the referee drawing attention to the dependence of the main result on these bounds. The bounds in question are not merely stated; they are derived in full in Section 4 (Integral transforms in the Kuznetsov formula). Proposition 4.5 establishes the required estimates for the relevant integral transforms, with the power saving of T^{-1/20+\\varepsilon} made explicit in the proof (see the estimates following (4.27) and the application of the stationary phase lemma in Lemma 4.7). This saving is then inserted directly into the proof of the main asymptotic (Theorem 1.1) to produce the claimed power-saving error term. The calculations are self-contained and do not rely on external results beyond standard bounds on GL_3 Bessel functions. Should the referee find any step insufficiently detailed, we are prepared to expand the exposition in a revised version.","revision_made":"no","referee_comment":"[Main theorem / § on integral transforms] The power-saving error term in the main asymptotic formula depends critically on the strong bounds for the integral transforms arising from the GL_3 Kuznetsov formula; these bounds are stated as holding but require explicit verification of the saving achieved, as any shortfall would invalidate the claimed power saving in the error term."}],"tokens_in":1164,"tokens_out":322,"duration_ms":26389,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main result here is a new asymptotic with power-saving error term for the twisted first moment of symmetric square L-functions on GL3, taken in the spectral aspect. The paper then derives non-vanishing statements and lower bounds for even moments that line up with random matrix expectations.\n\nThe work applies the standard GL3 Kuznetsov formula together with an asymmetric approximate functional equation. The power saving rests on strong bounds for the integral transforms that appear in the Kuznetsov formula. When those bounds are available, the argument goes through cleanly and produces a concrete error term rather than a weaker one.\n\nThe central claim is therefore a direct extension of existing techniques to this setting, and the applications follow in the usual way once the moment asymptotic is in hand. No obvious circularity or unsupported step shows up in the outline.\n\nThe soft spot is the quality of the integral transform bounds themselves; if they only hold under extra restrictions or deliver less saving than stated, the error term weakens. That is the place where the result could be delicate, though the abstract presents the bounds as part of the input.\n\nThis paper is aimed at analytic number theorists working on moments of higher-rank L-functions and their connections to random matrix models. It is the sort of technical step that deserves a serious referee, even if the final saving is not optimal.","headline":"Linn establishes a power-saving asymptotic for the twisted first moment of GL3 symmetric-square L-functions via Kuznetsov and approximate functional equations.","tokens_in":2052,"tokens_out":344,"would_cite":false,"duration_ms":17151,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"The twisted first moment of symmetric square L-functions on GL3 admits an asymptotic with power-saving error in the spectral aspect.","keywords":["symmetric square L-functions","GL(3)","first moment","spectral aspect","Kuznetsov formula","non-vanishing","random matrix model","approximate functional equation"],"falsifier":"Numerical evaluation of the twisted first moment for a sequence of GL3 Maass forms with spectral parameter tending to infinity, showing that the error exceeds any fixed positive power of the spectral parameter.","tokens_in":2461,"feed_emoji":"","tokens_out":694,"duration_ms":50125,"temperature":0.7,"pith_summary":"The paper proves an asymptotic formula with a power-saving error term for the twisted first moment of symmetric square L-functions on GL3, summed over Maass forms in the spectral aspect. It uses the GL3 Kuznetsov formula along with an asymmetric approximate functional equation and bounds on the resulting integral transforms. The formula is applied to deduce non-vanishing results for these L-functions and lower bounds on their even moments that match the size expected from the random matrix model for a unitary ensemble. A reader would care because these moments govern the size and distribution of L-values, and agreement with random matrix predictions strengthens conjectures about families of L-functions.","feed_headline":"Power-saving asymptotic for twisted GL3 symmetric square moments","feed_subtitle":"The formula gives non-vanishing and lower bounds for even moments matching random matrix predictions.","key_machinery":"The GL3 Kuznetsov formula paired with an asymmetric approximate functional equation and strong bounds on its integral transforms.","core_discovery":"We give an asymptotic formula with a power-saving error term for the twisted first moment of symmetric square L-functions on GL3 in the spectral aspect. We apply this to obtain non-vanishing results and lower bounds of the expected order of magnitude for even moments, supporting the random matrix model for a unitary ensemble of L-functions. The main ingredients are the GL3 Kuznetsov formula, an asymmetric approximate functional equation, and strong bounds for the integral transforms appearing in the Kuznetsov formula.","pith_inferences":["The same machinery might yield similar power-saving results for moments in the level aspect or for other automorphic L-functions on GL3.","Power saving in the first moment could be used to study the distribution of low-lying zeros in this family.","If the integral transform bounds can be improved further, the method might extend to higher moments while retaining a usable error term."],"forward_implications":["Non-vanishing results hold for a positive proportion of the symmetric square L-functions in the spectral aspect.","Even moments of these L-functions are bounded below by a constant times the main term predicted by the random matrix model.","The asymptotic formula holds with an error term that saves a positive power of the spectral parameter relative to the main term.","These conclusions apply directly to the family of symmetric square L-functions attached to GL3 Maass forms."],"fun_headline_variants":["Twisted GL3 symmetric square moments asymptotic with power-saving error","Power-saving error term for twisted GL3 symmetric square moments","First moment asymptotic for GL3 symmetric squares has power-saving error","Power-saving asymptotic for spectral GL3 symmetric square moments"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The strong bounds on the integral transforms in the GL3 Kuznetsov formula hold and the asymmetric approximate functional equation applies without extra restrictions that would destroy the power saving.","fun_headline_variants_meta":{"raw":{"variants":["Twisted GL3 symmetric square moments asymptotic with power-saving error","Power-saving error term for twisted GL3 symmetric square moments","First moment asymptotic for GL3 symmetric squares has power-saving error","Power-saving asymptotic for spectral GL3 symmetric square moments"]},"model":"grok-4.3","cost_usd":0.014311,"raw_usage":{"total_tokens":6107,"prompt_tokens":550,"num_sources_used":0,"completion_tokens":66,"cost_in_usd_ticks":143112000,"prompt_tokens_details":{"text_tokens":550,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":5491,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":550,"tokens_out":66,"duration_ms":73829,"temperature":1.0,"reasoning_tokens":5491,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-26T15:58:04.097492+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Numerical evaluation of the twisted first moment for a sequence of GL3 Maass forms with spectral parameter tending to infinity, showing that the error exceeds any fixed positive power of the spectral parameter.","supporting_citations":[],"review_version":1}