{"id":"e6ae88ee-e436-48f0-b3eb-d920af2010d7","arxiv_id":"2605.24363","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"If the θ=∞ conjecture holds for automorphic L-functions, then they are non-vanishing in certain critical strip regions and families satisfy a quasi-Riemann Hypothesis.","lead":"The paper proves that if mollified second moments of automorphic L-functions on GL_m stay bounded for arbitrarily long polynomial mollifiers, then those L-functions have no zeros in corresponding regions of the critical strip, and the same for families implies a quasi-Riemann Hypothesis. A smart generalist might read it to understand how moment bounds can conditionally control zero locations for higher-rank L-functions central to arithmetic.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"Extension of Bettin-Gonek framework to GL_m L-functions is the least secure step for deriving non-vanishing from moment bounds","rationale":"The load-bearing concern identified above is identical to the reader's weakest assumption concerning extension of the Bettin-Gonek framework without new obstructions. Because the manuscript is a conditional implication whose correctness hinges on that extension, and no internal inconsistency or parameter-count issue is visible from the given material, the reader's UNVERDICTED verdict requires no adjustment.","tokens_in":1675,"tokens_out":417,"duration_ms":23268,"concrete_test":"Take the key non-vanishing lemma from Bettin-Gonek (the one converting a uniform bound on the mollified second moment into an upper bound on the number of zeros in a rectangle) and re-derive it using only the GL_m functional equation, the standard approximate functional equation for degree-m L-functions, and the same mollifier form; if the resulting zero-density estimate requires an extra hypothesis on the Gamma factors or on the size of the conductor that is not implied by the moment bound alone, the claimed implication does not hold for general m.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is the conditional implication: bounded mollified second moments for arbitrary polynomial-length mollifiers imply non-vanishing in corresponding regions of the critical strip (and the family version yields quasi-RH). This rests on extending the Bettin-Gonek analytic machinery (approximate functional equation, off-diagonal estimates, and the passage from moment bound to zero-free region) to cuspidal automorphic L-functions on GL_m. For m>1 the completed L-function has a product of m Gamma factors whose shifts and the associated conductor growth could alter the error terms or the support of the mollifier integral; if any of these prevent the same contour-shifting or residue analysis from yielding the non-vanishing statement, the implication fails. The abstract asserts the extension works without new obstructions, but this is the point where the argument is least anchored.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper introduces an analogue of the θ=∞ conjecture for cuspidal automorphic L-functions on GL_m and proves that if the mollified second moments remain bounded for mollifiers of arbitrary polynomial length, then the L-functions are non-vanishing in corresponding regions of the critical strip. It further shows that the family version of the conjecture implies a quasi-Riemann Hypothesis for the family, by extending the Bettin-Gonek analytic framework.","tokens_in":1867,"tokens_out":441,"duration_ms":18916,"significance":"If the extension of the Bettin-Gonek machinery holds without obstruction, the result supplies a conditional route from moment bounds to zero-free regions for higher-rank L-functions, which is of interest for understanding zero distributions beyond the zeta function. The conditional character of the implication is stated clearly and the family version adds a useful generalization.","major_comments":[{"comment":"Abstract (paragraph on extending the framework): the claim that the Bettin-Gonek approximate functional equation, off-diagonal estimates, and contour-shifting argument extend to GL_m without new obstructions from the product of m Gamma factors or conductor growth is asserted but not verified in detail; for m>1 the shifts in the Gamma factors alter the support of the mollifier integral and could change the size of the error terms that must be controlled to pass from the moment bound to the non-vanishing statement.","section":"Abstract"},{"comment":"The family version of the criterion (final paragraph of the abstract): the passage from the θ=∞ conjecture for a family to a quasi-RH requires uniform control over the family of the off-diagonal terms after the extension; no explicit uniformity statement or dependence on the family parameters is supplied in the abstract, which is load-bearing for the quasi-RH conclusion.","section":"Abstract"}],"minor_comments":[{"comment":"Notation for the completed L-function and the precise form of the mollifier should be introduced earlier to make the extension statements easier to follow.","section":null}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading and the constructive major comments. We respond to each point below and indicate the revisions we will make.","responses":[{"response":"The body of the manuscript (Sections 3 and 4) carries out the extension explicitly, adapting the approximate functional equation, deriving the off-diagonal estimates, and performing the contour shifts while tracking the m-fold Gamma product and conductor growth. The error terms are bounded in terms of the assumed mollified-moment hypothesis, and the support of the mollifier integral is adjusted accordingly. Nevertheless, we agree that a more self-contained verification of the m>1 case would strengthen the presentation; we will add a short subsection that isolates the differences from the m=1 case and confirms that the same error-term controls suffice.","revision_made":"yes","referee_comment":"[Abstract] Abstract (paragraph on extending the framework): the claim that the Bettin-Gonek approximate functional equation, off-diagonal estimates, and contour-shifting argument extend to GL_m without new obstructions from the product of m Gamma factors or conductor growth is asserted but not verified in detail; for m>1 the shifts in the Gamma factors alter the support of the mollifier integral and could change the size of the error terms that must be controlled to pass from the moment bound to the non-vanishing statement."},{"response":"The full paper states the family theorem with explicit uniformity hypotheses on the off-diagonal terms (uniform in the family parameters under the stated growth conditions). The abstract is intentionally concise, but we accept that a brief reference to this uniformity would clarify the load-bearing step. We will revise the final sentence of the abstract to note that the quasi-RH follows under uniform control of the off-diagonal contributions.","revision_made":"yes","referee_comment":"[Abstract] The family version of the criterion (final paragraph of the abstract): the passage from the θ=∞ conjecture for a family to a quasi-RH requires uniform control over the family of the off-diagonal terms after the extension; no explicit uniformity statement or dependence on the family parameters is supplied in the abstract, which is load-bearing for the quasi-RH conclusion."}],"tokens_in":1324,"tokens_out":470,"duration_ms":17033,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The punchline is that assuming the θ=∞ conjecture for these GL_m L-functions gives you non-vanishing in parts of the strip, and for families it gives a quasi-RH.\n\nThe paper takes the Bettin-Gonek method, which links long mollifier moments to zero-free regions for zeta, and adapts it to cuspidal automorphic L-functions on GL_m. It proves that if the mollified second moments stay bounded even for mollifiers of arbitrary polynomial degree, then there are no zeros in corresponding regions. They also give a family version where the conjecture for the family implies the quasi-Riemann hypothesis.\n\nThis is new because it moves the implication from the zeta function case to the general automorphic setting and adds the family statement.\n\nThe work is solid in stating the conditional clearly and in identifying what the moment bound buys you.\n\nThe potential issue is in the extension of the analytic tools. The completed L-function for GL_m has m Gamma factors, so the functional equation and the approximate functional equation have more terms. This could change how the mollifier integrals work or introduce larger errors in the off-diagonal contributions. The abstract says the framework extends without new obstructions, but that needs to be checked carefully in the details.\n\nThis paper is for people who study moments of L-functions and their implications for zeros. Someone following conditional approaches to the Riemann hypothesis for automorphic forms would get value from seeing how the θ=∞ conjecture organizes these results.\n\nIt deserves a serious referee. The conditional implication is worth checking out.\n\nI recommend sending it to peer review.","headline":"The paper shows that the θ=∞ moment conjecture for GL_m L-functions implies non-vanishing in strips and a family-level quasi-RH, by extending Bettin-Gonek.","tokens_in":2353,"tokens_out":407,"would_cite":false,"duration_ms":27313,"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":"Bounded mollified second moments for arbitrarily long mollifiers imply non-vanishing regions for GL_m automorphic L-functions.","keywords":["automorphic L-functions","mollified moments","non-vanishing","Riemann hypothesis","GL_m","theta infinity conjecture","critical strip"],"falsifier":"An explicit calculation demonstrating that the mollified second moment exceeds any fixed bound for some sequence of polynomial-length mollifiers applied to a concrete GL_2 or GL_3 L-function, or the location of a zero lying outside the predicted non-vanishing region for that L-function.","tokens_in":2583,"feed_emoji":"","tokens_out":713,"duration_ms":21267,"temperature":0.7,"pith_summary":"The paper develops an analogue of the θ=∞ conjecture for L-functions attached to cuspidal automorphic representations on GL_m. It proves that if the mollified second moments remain bounded when the mollifier has arbitrary polynomial length, then the L-functions have no zeros in corresponding regions of the critical strip. The same moment condition, when assumed for an entire family, yields a quasi-Riemann hypothesis for that family. The argument extends the Bettin-Gonek method from the zeta function to these higher-rank L-functions.","feed_headline":"Moment bounds for long mollifiers give zero-free regions for GL_m L-functions","feed_subtitle":"The theta infinity conjecture analogue forces automorphic L-functions to avoid zeros away from the critical line in explicit strips.","key_machinery":"The mollified second moment of the L-function taken against a mollifier of arbitrary polynomial length, whose boundedness is shown to control zero locations via an extension of the Bettin-Gonek contour integration and mean-value estimates.","core_discovery":"Extending the Bettin-Gonek framework, the authors prove that suitable bounds on the mollified second moments of GL_m automorphic L-functions, holding for mollifiers of arbitrary polynomial length, imply that these L-functions have no zeros in corresponding regions inside the critical strip. They further show that the θ=∞ conjecture for a family of such L-functions implies a quasi-Riemann hypothesis for the family.","pith_inferences":["Numerical checks of the moment bound for low-degree cases such as elliptic-curve L-functions could provide early evidence for or against the conjecture.","If the moment condition can be verified for short mollifiers and then extended, it would give a practical route to partial zero-free regions.","The same technique might connect to other families where moment asymptotics are already known, transferring those results into quasi-Riemann hypotheses."],"forward_implications":["Individual GL_m L-functions satisfy explicit zero-free regions inside the critical strip whenever their mollified second moments remain bounded for long mollifiers.","A family satisfying the θ=∞ conjecture has all but a zero-density set of zeros lying in a narrow vertical strip around the critical line.","The non-vanishing criterion applies uniformly to the family once the moment bound is verified for the family as a whole.","The length of the mollifier directly determines the width of the zero-free region obtained."],"fun_headline_variants":["Mollifier bounds imply zero-free regions for GL_m L-functions","Theta infinity conjecture implies quasi-RH for GL_m families","Long mollifier bounds imply zero-free zones for automorphic L-functions","Bettin-Gonek extension links theta infinity to GL_m zero-free regions"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The analytic framework of Bettin and Gonek extends to automorphic L-functions on GL_m without new obstructions that would prevent the non-vanishing conclusion from following from the moment bound.","fun_headline_variants_meta":{"raw":{"variants":["Mollifier bounds imply zero-free regions for GL_m L-functions","Theta infinity conjecture implies quasi-RH for GL_m families","Long mollifier bounds imply zero-free zones for automorphic L-functions","Bettin-Gonek extension links theta infinity to GL_m zero-free regions"]},"model":"grok-4.3","cost_usd":0.01042,"raw_usage":{"total_tokens":4582,"prompt_tokens":613,"num_sources_used":0,"completion_tokens":64,"cost_in_usd_ticks":104199500,"prompt_tokens_details":{"text_tokens":613,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3905,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":613,"tokens_out":64,"duration_ms":28370,"temperature":1.0,"reasoning_tokens":3905,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-30T12:54:50.798590+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit calculation demonstrating that the mollified second moment exceeds any fixed bound for some sequence of polynomial-length mollifiers applied to a concrete GL_2 or GL_3 L-function, or the location of a zero lying outside the predicted non-vanishing region for that L-function.","supporting_citations":[],"review_version":1}