{"id":"4b5f5e5a-1b8e-451d-95ae-50aa4fd55466","arxiv_id":"2607.00282","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"At least 1/9 of zeros of L(s, Π₀ × χ) lie on the critical line for large Q, with new power-saving mean square asymptotics for PGL(3) twists (unconditional if self-dual) and stronger results for PGL(2).","lead":"The paper proves that at least 1/9 of the zeros of L(s, Π₀ × χ) for cuspidal automorphic representations Π₀ of PGL(3) lie on the critical line as the conductor Q goes to infinity. A smart generalist might read it because it advances tools for studying zero distributions of L-functions, which connect to prime number behavior and arithmetic conjectures.","discovery_kind":"unclear","skeptic_critique":{"model":"grok-4.3","headline":"Mean-value theorem only for T ≪ Q^{1/3-ε} controls zeros in a vanishing proportion of the total (conductor ≍ Q^3)","rationale":"The reader identified the asymptotic formula itself as the weakest point. The actual load-bearing gap is narrower but more decisive: even granting the formula, its stated range T ≤ Q^{1/3-ε} is too short to reach a positive proportion of all zeros, given the conductor growth.","tokens_in":1982,"tokens_out":464,"duration_ms":62540,"concrete_test":"Locate the precise statement of the main theorem asserting the 1/9 proportion (likely Thm. 1.1 or 1.2). Check whether it explicitly restricts to |t| ≤ Q^{1/3} or claims the proportion over all zeros. If the former, compute the ratio of zero counts up to height Q^{1/3} versus height Q^3 using the standard density (T/2π) log(cond) to verify the ratio is o(1).","verdict_should_be":"REJECT","load_bearing_attack":"The central claim is that ≥1/9 of all zeros of L(s, Π₀ × χ) lie on Re(s)=1/2 as Q→∞. Levinson's method is applied using the new mean-square asymptotic, which is stated to hold only for Q^ε ≤ T ≤ Q^{1/3-ε} (with Dirichlet polynomials of length <1/2-ε when T=Q^ε). For fixed Π₀ on PGL(3), the analytic conductor of L(s, Π₀ × χ) is ≍ Q^3. The number of zeros with |Im ρ| ≪ X is ~ X log(Q^3). Setting X = Q^{1/3} therefore captures only a Q^{-8/3+o(1)} fraction of the zeros with |Im ρ| ≪ Q^3. The method therefore yields at best a positive proportion among the low-height zeros, which is o(1) of the full set. The abstract states the 1/9 proportion without a height restriction, so the range limitation prevents the claimed conclusion from following.","agreement_with_reader":"disagree"},"referee_report":{"model":"grok-4.3","summary":"The paper claims that as Q→∞, at least 1/9 of the zeros of L(s, Π₀ × χ) lie on the critical line, where Π₀ is a cuspidal automorphic representation of PGL(3, A_Q) and χ ranges over primitive Dirichlet characters of conductor ≤Q. The result is unconditional when Π₀ is self-dual (and holds under a mild condition otherwise). Analogous, quantitatively stronger results are obtained for PGL(2). The proof applies Levinson's method to a new asymptotic mean-square formula for L(s, Π₀ × χ) times an arbitrary Dirichlet polynomial, valid with power-saving error in the range Q^ε ≤ T ≤ Q^{1/3-ε} (with length restrictions when T=Q^ε). The work also develops a refined asymptotic large sieve without Ramanujan assumptions and provides evidence for CFKRS conjectures.","tokens_in":2218,"tokens_out":509,"duration_ms":23740,"significance":"If the central claim holds, the result would constitute a notable advance by establishing the first unconditional positive-proportion theorems on critical zeros for twisted PGL(3) L-functions in large families, extending Levinson-type results beyond GL(2). The new mean-value theorem, uniform in both T and Q aspects and free of Ramanujan hypotheses, is a flexible technical tool with potential applicability to other analytic-number-theory problems. The explicit use of Hecke-algebra computations and the unconditional PGL(2) case are additional strengths.","major_comments":[{"comment":"Abstract and paragraph on key technical input: the mean-square asymptotic is stated only for Q^ε ≤ T ≤ Q^{1/3-ε}. For fixed Π₀ on PGL(3) the analytic conductor is ≍ Q^3, so the total number of zeros with |Im ρ| ≪ Q^3 is ~ Q^3 log Q. The range T ≪ Q^{1/3} therefore controls only a Q^{-8/3+o(1)} fraction of all zeros. Levinson's method applied in this range yields at best a positive proportion among low-height zeros, which is o(1) of the full set; the manuscript must explain how the claimed 1/9 proportion of ALL zeros (without height restriction) follows.","section":"Abstract and key technical input paragraph"}],"minor_comments":[],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the detailed and constructive report. The major comment raises an important point about the precise scope of our main theorem, which we address below by agreeing to revise the manuscript for clarity.","responses":[{"response":"We agree that the abstract and introduction as currently worded do not explicitly restrict attention to zeros of bounded height, which risks misinterpretation. Our Levinson-type argument, relying on the new mean-square asymptotic valid only for T ≪ Q^{1/3-ε}, in fact establishes that a positive proportion (at least 1/9) of the zeros lying in the range |Im ρ| ≪ Q^{1/3-ε} are on the critical line. This is still a new unconditional result for the PGL(3) family in the low-height regime. We will revise the abstract, the statement of the main theorem, and the discussion of the key technical input to make the height restriction explicit and to add a clarifying remark on the range of applicability. The PGL(2) results will be treated analogously where appropriate.","revision_made":"yes","referee_comment":"[Abstract and key technical input paragraph] Abstract and paragraph on key technical input: the mean-square asymptotic is stated only for Q^ε ≤ T ≤ Q^{1/3-ε}. For fixed Π₀ on PGL(3) the analytic conductor is ≍ Q^3, so the total number of zeros with |Im ρ| ≪ Q^3 is ~ Q^3 log Q. The range T ≪ Q^{1/3} therefore controls only a Q^{-8/3+o(1)} fraction of all zeros. Levinson's method applied in this range yields at best a positive proportion among low-height zeros, which is o(1) of the full set; the manuscript must explain how the claimed 1/9 proportion of ALL zeros (without height restriction) follows."}],"tokens_in":1719,"tokens_out":418,"duration_ms":43140,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The headline result does not hold as stated. The new asymptotic for the mean square of L(s, Π₀ × χ) times a Dirichlet polynomial is only established for Q^ε ≤ T ≤ Q^{1/3-ε}. For fixed Π₀ on PGL(3) the analytic conductor is ~Q^3, so the total zeros up to height ~Q^3 number roughly Q^3 log Q. The T-range only reaches height Q^{1/3}, which captures a vanishing fraction Q^{-8/3+o(1)} of them. Levinson's method therefore yields a positive proportion only among those low zeros, not among all zeros as Q→∞. The abstract states the 1/9 figure without height restriction, so the central claim needs either a larger T-range or a corrected statement.\n\nWhat is actually new is the power-saving asymptotic itself in the T/Q aspects, allowing Dirichlet polynomials of length <1/2-ε when T is small and giving a strong error term O(Q^{7/4+ε}) when the polynomial is constant. The argument avoids any Ramanujan hypothesis and uses the arithmetic of the fixed Π₀ together with Mathematica to manage the Hecke relations. The PGL(2) results are stronger, fully unconditional, and appear to be new as well. The work also supplies evidence toward the CFKRS conjectures in this setting.\n\nThe derivation of the asymptotic is the main technical load and cannot be checked from the abstract alone, but the setup looks consistent on its own terms. The mild extra condition for non-self-dual Π₀ is left vague in the summary. No circularity is visible.\n\nThis paper is for specialists in analytic number theory who care about mean-value theorems and zero statistics for higher-rank L-functions. The mean-value result may be reusable even if the zero proportion needs repair. It deserves a serious referee to verify the asymptotic and to decide whether the zero claim can be salvaged or must be restated as a result on low-lying zeros.","headline":"The claimed 1/9 proportion of critical zeros for PGL(3) twists does not follow from the stated range of the new mean-value asymptotic.","tokens_in":2747,"tokens_out":488,"would_cite":false,"duration_ms":22694,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M26","11F66"],"pacs":[],"model":"grok-4.3","headline":"At least one ninth of the zeros of L(s, Π₀ × χ) lie on the critical line as the conductor Q tends to infinity.","keywords":["L-functions","critical zeros","Levinson method","mean value theorems","automorphic representations","Dirichlet characters","PGL(2)","PGL(3)"],"falsifier":"An explicit numerical check or theoretical construction showing that the proportion of critical zeros falls below 1/9 for some fixed self-dual Π₀ and a sequence of arbitrarily large Q would disprove the main claim.","tokens_in":2885,"feed_emoji":"","tokens_out":851,"duration_ms":30526,"temperature":0.7,"pith_summary":"The paper proves that in the family of L-functions attached to a fixed cuspidal automorphic representation Π₀ of PGL(3) twisted by primitive Dirichlet characters χ of conductor up to Q, at least 1/9 of the zeros lie on the critical line when Q is large. The argument uses Levinson's method and rests on a new mean-square asymptotic for L(s, Π₀ × χ) times an arbitrary Dirichlet polynomial that saves a positive power of Q uniformly in the range Q^ε ≤ T ≤ Q^{1/3-ε}. The result holds unconditionally when Π₀ is self-dual and under a mild extra hypothesis otherwise; the corresponding statements for PGL(2) are fully unconditional and quantitatively stronger. The same estimates supply evidence for the CFKRS conjectures at large height and large twist.","feed_headline":"At least 1/9 of zeros lie on critical line for twisted PGL L-functions","feed_subtitle":"New mean-square estimates let Levinson's method produce an explicit positive proportion unconditionally when the form is self-dual.","key_machinery":"A new asymptotic formula with power-saving error term for the mean square of L(s, Π₀ × χ) times an arbitrary Dirichlet polynomial, valid uniformly in both the T- and Q-aspects for Q^ε ≤ T ≤ Q^{1/3-ε}.","core_discovery":"As Q tends to infinity, at least 1/9 of the zeros of L(s, Π₀ × χ) lie on the critical line, where Π₀ is a cuspidal automorphic representation of PGL(3,A_Q) and χ runs over primitive Dirichlet characters of conductor ≤ Q. The result is unconditional for self-dual Π₀ and holds under a mild condition otherwise. For representations of PGL(2) the statements are fully unconditional and give a stronger proportion. The proof relies on a new power-saving asymptotic for the mean square of L(s, Π₀ × χ) times an arbitrary Dirichlet polynomial valid for T up to Q^{1/3-ε}.","pith_inferences":["If similar mean-square asymptotics can be proved for higher-rank groups, the Levinson method would give positive proportions of critical zeros in those families as well.","Improving the error term in the mean square to allow longer Dirichlet polynomials might raise the 1/9 proportion.","The uniformity in both T and Q suggests the estimates could be used to study zeros near the edge of the critical strip in other twist families."],"forward_implications":["The same mean-value theorem yields the result for all self-dual Π₀ of PGL(3) without further hypotheses.","For PGL(2) the method produces a larger explicit proportion and applies without any extra conditions.","The estimates furnish evidence for the CFKRS conjectures when both the twist and the height are large.","The mean-value theorem is stated to be applicable to other problems in analytic number theory."],"fun_headline_variants":["1/9 of zeros on critical line for twisted PGL L-functions","Critical zeros for 1/9 of PGL(3) twists when self-dual","Stronger proportion for PGL(2) critical zeros unconditional","Power-saving mean squares for PGL(2) and PGL(3) twists"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The new mean-square asymptotic for the product of the twisted L-function and an arbitrary Dirichlet polynomial holds with a power-saving error term throughout the stated range of T and Q.","fun_headline_variants_meta":{"raw":{"variants":["1/9 of zeros on critical line for twisted PGL L-functions","Critical zeros for 1/9 of PGL(3) twists when self-dual","Stronger proportion for PGL(2) critical zeros unconditional","Power-saving mean squares for PGL(2) and PGL(3) twists"]},"model":"grok-4.3","cost_usd":0.010505,"raw_usage":{"total_tokens":4769,"prompt_tokens":919,"num_sources_used":0,"completion_tokens":82,"cost_in_usd_ticks":105049500,"prompt_tokens_details":{"text_tokens":919,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3768,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":919,"tokens_out":82,"duration_ms":36821,"temperature":1.0,"reasoning_tokens":3768,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-02T01:21:35.453863+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit numerical check or theoretical construction showing that the proportion of critical zeros falls below 1/9 for some fixed self-dual Π₀ and a sequence of arbitrarily large Q would disprove the main claim.","supporting_citations":[],"review_version":1}