{"id":"2d0770a0-c98f-4b17-b718-8a708a4f4bf3","arxiv_id":"2508.14888","paper_version":2,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":8.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors establish large sieve inequalities for GL_n automorphic L-functions that are independent of the Ramanujan conjecture and improve short sums, with applications to moments and zero density estimates.","lead":"This paper proves new large sieve inequalities for families of automorphic L-functions on GL_n, removing prior assumptions about the generalized Ramanujan conjecture and giving the first improvement over the trivial bound for short sums. The results produce stronger moment bounds and zero density estimates that appear to be the best known for arbitrary families.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No specific flaw identifiable from the abstract; chief risk is conductor uniformity for ramified local factors in the π×π0 family.","rationale":"Neither the full proof nor even a full theorem statement is available, so no internal inconsistency can be definitively located. The central claim is a quantitative large sieve inequality for automorphic coefficients, which is a known and plausible type of result for GL_n; the abstract's applications are natural consequences. The weakest point, in agreement with the reader, is the conductor-uniform treatment of local factors—especially ramified ones in the Rankin-Selberg and log-L contexts. My concern is more specific: without a stated conductor parameter and the exact range of S,N, the 'improvement over trivial' claim is not independently checkable. That said, this is a risk rather than a demonstrated error, so I do not move the reader's already-UNVERDICTED verdict.","tokens_in":850,"tokens_out":15911,"duration_ms":227557,"concrete_test":"Obtain Theorem 1 from the full text. Specialize to F=Q, n=n0=1: with S the set of all Dirichlet characters of conductor ≤Q and π0 trivial, the inequality must become the classical large sieve ∑_{q≤Q}∑_{χ(q)} |∑_{n≤N} a_n χ(n)|^2 ≪ (N+Q)∑|a_n|^2. Then take one fixed ramified π0 (e.g., a supercuspidal of conductor p^m on GL_2(Q_p)) and a family S of unramified π of conductor ≤Q; verify that the displayed bound for the π×π0 coefficients has no factor depending on p^m beyond the analytic conductor C(π×π0). If either check fails, the advertised short-sum improvement and independence from GRC do not follow.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The abstract asserts a large sieve inequality for arbitrary finite S⊂F_n, for both L(s,π) and L(s,π×π0), handling L, L^{-1}, and log L, with no unproved Ramanujan input, plus a first short-sum improvement. This is a strong but plausible extension of known analytic-conductor large sieves. The load-bearing unstated premise is that the local factors at primes ramified in π or π0—especially in the Rankin-Selberg twist—admit uniform, conductor-weighted bounds that do not use GRC. The coefficients of L^{-1} and log L at such primes are not simple Hecke eigenvalues; they come from local zeta integrals and their size can depend on the conductor exponent in a way that the analytic-conductor parameter may not fully expose. If the proof bounds these local coefficients by a crude p^{fθ} with f the conductor exponent, the saved factor in the large sieve may be absorbed by ramified locations and the improvement over the trivial Ramanujan-dependent bound disappears. The abstract does not state the theorem's hypotheses, conductor parameter, or the regime of |S| and N in which 'improves upon the trivial bound' holds, so this cannot be independently checked.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper announces large sieve inequalities for the families {L(s,π): π∈S} and {L(s,π×π0): π∈S}, where S is an arbitrary finite subset of unitary cuspidal automorphic representations of GL_n over a number field. The stated results are claimed to be independent of progress toward the generalized Ramanujan conjecture and to treat simultaneously the Dirichlet coefficients of L, L^{-1}, and log L. The abstract also claims the first such result that improves on the trivial bound for short sums, with applications to second moments at the central point, zero-density estimates near Re(s)=1, unconditional log-free zero-density estimates for Rankin-Selberg families, and a density theorem for non-archimedean Langlands parameters.","tokens_in":1169,"tokens_out":2097,"duration_ms":26806,"significance":"If the announced results are correct, they constitute a substantial advance: they would remove a major unproved analytic input (Ramanujan-type bounds) from a general large-sieve machinery for automorphic L-functions, extend the scope to Rankin-Selberg twists, and yield new unconditional or stronger applications. The proof is not verifiable from the abstract alone; the plausibility rests on the authors' prior record and on the apparent continuity with existing analytic-conductor large sieves. The paper would be of high interest to analytic number theorists working on automorphic L-functions, moments, and zero-density estimates.","major_comments":[{"comment":"The main theorem is stated only in prose; the precise hypotheses are missing. In particular, for the large sieve inequalities to be meaningful one needs to know the analytic conductor parameter Q(S), the dependence of the implied constants on n, n0, the number field F, and the allowable range of |S| and N. Without these hypotheses and quantifications, the central claim 'independent of progress towards the generalized Ramanujan conjecture' cannot be independently checked. The full manuscript must state the exact form of the inequality, including the power of Q and the admissible ranges.","section":"Abstract, first paragraph"},{"comment":"For the families L(s,π) and especially L(s,π×π0), the coefficients of L^{-1} and log L at primes ramified in π or π0 are not simple Hecke eigenvalues; they derive from local zeta integrals and may depend on the conductor exponent. A load-bearing point is whether these coefficients satisfy uniform bounds that do not use the generalized Ramanujan conjecture. If the proof bounds them by a crude p^{fθ} with f the local conductor exponent, the saving supplied by the large sieve could be absorbed by ramified places, and the claimed improvement over the trivial Ramanujan-dependent bound could fail. The manuscript should explicitly display the local coefficient bounds and show that the conductor-weighting in the sieve is preserved.","section":"Abstract, first paragraph (local factors at ramified primes)"},{"comment":"The claim that this is 'the first such result that improves upon the trivial bound for short sums' is not assessable from the abstract because the term 'short sums' is not defined, nor is the comparison with the trivial bound stated. To verify this claim, the paper must specify the length N relative to the analytic conductor Q and the size of S for which the improvement holds, and give the resulting exponent beyond the trivial bound. The abstract currently leaves open the possibility that the improvement occurs only in a narrow parameter regime.","section":"Abstract, short-sum improvement"},{"comment":"The applications listed are presented without quantitative statements. Since the entire evaluation of the paper's contribution depends on the strength of these consequences, the manuscript should provide precise statements for at least the key applications: the exact bound for ∑|L(1/2,π)|^2 for arbitrary S, the zero-density exponent in the GRH-violation estimate, and the unconditional log-free zero-density estimate. Without these, the significance of the results relative to Brumley-Thorner-Zaman and Lichtman-Pascadi cannot be measured.","section":"Abstract, applications"}],"minor_comments":[{"comment":"The abstract would be more useful if it identified theorem numbers corresponding to each announced result and stated the main analytic conductor parameter explicitly. This is a presentation issue, but it would aid the reader substantially.","section":"Abstract, general"}],"recommendation":"uncertain","confidential_remarks":"I have reviewed only the abstract; the full text was not available. The announced results are plausible and important if correct, but the central technical points—especially the treatment of ramified local factors for L^{-1} and log L, and the precise short-sum regime—cannot be checked from the abstract. My recommendation is 'uncertain' pending inspection of the full manuscript. I saw no obvious sign of circularity in the abstract; the results are framed as extensions of prior work, which is appropriate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The abstract promises something genuinely new: a GL_n large sieve that does not depend on progress toward Ramanujan, handles L, L^{-1}, and log L simultaneously, and gives the first short-sum improvement over the trivial bound. That is a meaningful step beyond Brumley–Thorner–Zaman and Lichtman–Pascadi. The applications are concrete and, if they all follow from the stated inequality, they are strong. The authors are well placed to do this, and the abstract reads like a sober claim of substance rather than a sales pitch.\n\nThat said, we only have the abstract, so the proof is the whole ballgame. My chief concern is the same one you flagged: uniformity at ramified primes, especially in the π×π0 family. The Dirichlet coefficients of L^{-1} and log L are not Hecke eigenvalues; they come from local zeta integrals, and their size can depend on the conductor exponent in ways that an analytic conductor parameter might not fully control. If the proof bounds them crudely by p^{fθ}, the savings from dropping Ramanujan could be absorbed by ramified places and the short-sum improvement could evaporate. The abstract does not state the hypotheses on S, the conductor parameter, or the regime where the short-sum bound beats trivial, so I can't check that from here. This is not a demonstrated flaw—just the spot where I'd focus a referee effort.\n\nI also note that 'arbitrary S' is doing a lot of work. The large sieve constants presumably depend on the analytic conductors in some way, and without that dependence spelled out, I can't independently verify the claimed second moment bound. Again, not an accusation, just a limit of an abstract-only read.\n\nIf the proof is solid, this paper is for analytic number theorists working on automorphic L-functions and families. It deserves a serious referee even now; the claims are precise and falsifiable by reading the argument. I wouldn't cite it in my own work until I've seen the proof of the short-sum claim, but that's true of any paper this strong. My recommendation: send it out, and ask the referee to scrutinize the ramified local factors and the conductor dependence in the log L case. If those hold up, this is a clean, important contribution.","headline":"If the proof matches the abstract, this is a real advance in GL_n large sieves, but the ramified local coefficients for log L and L^{-1} are where I'd check first.","tokens_in":1572,"tokens_out":1672,"would_cite":false,"duration_ms":19053,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11F66","11F70","11M41"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper establishes large sieve inequalities for families of automorphic L-functions, including their inverse and logarithmic coefficients, without requiring any unproved progress toward the generalized Ramanujan conjecture, and gives th","keywords":["large sieve","automorphic L-functions","Rankin-Selberg","generalized Ramanujan conjecture","zero density","Dirichlet coefficients","moments of L-functions","GL_n"],"falsifier":"For a specific finite family S with known conductor distribution, evaluate both sides of the paper's large sieve inequality numerically; a single family for which the claimed inequality fails, or for which the short-sum bound is not below trivial, would refute the paper's main claim.","tokens_in":815,"feed_emoji":"🔢","tokens_out":4936,"duration_ms":53927,"temperature":0.7,"pith_summary":"This paper proves new large sieve inequalities for finite families of automorphic L-functions, both L(s,π) and their Rankin-Selberg products L(s,π×π0), over a number field. The inequalities are independent of the generalized Ramanujan conjecture, so they do not rely on unproved bounds for Hecke eigenvalues. They simultaneously handle the Dirichlet coefficients of L, of L^{-1}, and of log L, which earlier large sieves could not do. Because of this, the paper obtains the first large sieve of this kind that beats the trivial bound for short sums of coefficients. If correct, these inequalities yield the strongest known unconditional bound for the second moment of automorphic L-functions at the critical point, as well as sharper zero-density estimates near Re(s)=1.","feed_headline":"Large sieve for GL_n L-functions works without Ramanujan bounds","feed_subtitle":"The sieve handles L, L^{-1}, and log L coefficients and beats trivial bounds on short sums.","key_machinery":"The engine is a large sieve inequality: a bound on a sum over a family of L-functions of weighted Dirichlet coefficients, expressed in terms of the conductors of the family and the length of the coefficient sum. The new machinery obtains this inequality simultaneously for the coefficients of L(s,π), L(s,π)^{-1}, and log L(s,π), using the functional equation and Euler product structure of both L(s,π) and L(s,π×π0) in the conductor aspect. The key structural step is showing that the inverse and logarithmic coefficients can be controlled without the generalized Ramanujan conjecture, through uniformity in the conductor that avoids pointwise bounds on Hecke eigenvalues.","core_discovery":"The paper's central discovery is a large sieve inequality for an arbitrary finite family S of unitary cuspidal automorphic representations of GL_n over a number field. For the L-functions L(s,π) and the Rankin-Selberg products L(s,π×π0), the sieve bounds weighted sums of the Dirichlet coefficients of L, of L^{-1}, and of log L uniformly in the conductors, without invoking any unproved bound toward the generalized Ramanujan conjecture. It is the first large sieve of this type to improve on the trivial bound for short sums. The resulting inequalities give, for arbitrary S, the strongest known unconditional bound on Σ_{π∈S}|L(1/2,π)|^2, stronger zero-density estimates near Re(s)=1 for families","pith_inferences":["Because the sieve covers L^{-1} and log L coefficients, it should plug directly into mollifier constructions, which would yield unconditional non-vanishing or lower-bound results for moments of GL_n L-functions; the paper does not pursue that step.","The removal of the Ramanujan-conjecture dependence is a structural feature, suggesting the same conductor-aspect technique may transfer to other families of L-functions (for example symmetric powers or function-field analogues) whose analytic properties are known, though the paper does not claim such a transfer.","The short-sum improvement could be tested numerically on a concrete family, such as level-one holomorphic modular forms, where the Dirichlet coefficients are essentially divisor functions; a violation of the predicted improvement there would localize the limit of the method."],"forward_implications":["Unconditional moment bounds: for any finite family S, the inequality gives the strongest known bound on Σ_{π∈S}|L(1/2,π)|^2, with no Ramanujan-type hypothesis.","Sharper zero-density estimates: the number of possible violations of the generalized Riemann hypothesis in a thin strip near Re(s)=1 is counted more tightly for automorphic and Rankin-Selberg L-functions.","Hypothesis-free log-free zero density: the conditional log-free zero density estimate for families of Rankin-Selberg L-functions now holds without any unproven hypotheses.","Better counting of bad Langlands parameters: the density theorem for non-archimedean Langlands parameters is improved, making violations of the generalized Ramanujan conjecture known to be rarer."],"supporting_citations":[],"fun_headline_variants":["GL_n large sieve: no Ramanujan bounds needed","L-function sieve beats trivial short sums on GL_n","Large sieve for GL_n removes all unproven hypotheses","Strongest moment bound yet from GL_n large sieve","GL_n sieve improves zero density and moment estimates"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The argument assumes the standard analytic properties of the automorphic and Rankin-Selberg L-functions—functional equation and Euler product—hold uniformly in the conductor aspect, especially at ramified primes; a failure of that uniformity would break the sieve's control of the implied constants.","fun_headline_variants_meta":{"raw":{"variants":["GL_n large sieve: no Ramanujan bounds needed","L-function sieve beats trivial short sums on GL_n","Large sieve for GL_n removes all unproven hypotheses","Strongest moment bound yet from GL_n large sieve","GL_n sieve improves zero density and moment estimates"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000719,"raw_usage":{"total_tokens":3119,"prompt_tokens":853,"completion_tokens":2266,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":597,"completion_tokens_details":{"reasoning_tokens":2190}},"tokens_in":597,"tokens_out":2266,"duration_ms":18842,"temperature":1.0,"reasoning_tokens":2190,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T18:10:41.124172+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a specific finite family S with known conductor distribution, evaluate both sides of the paper's large sieve inequality numerically; a single family for which the claimed inequality fails, or for which the short-sum bound is not below trivial, would refute the paper's main claim.","supporting_citations":[],"review_version":1}