{"id":"cff9f709-919b-4cc6-ab58-b20b64872918","arxiv_id":"2606.31441","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"Under Artin's conjecture, an explicit asymptotic counts zeros of Artin L-functions up to height T, giving unconditional formulas for Hecke L-functions.","lead":"The paper derives an explicit asymptotic formula counting non-trivial zeros of Artin L-functions up to height T, assuming Artin's holomorphy conjecture. This yields unconditional zero-counting formulas for Hecke L-functions over number fields and refines earlier results on zeta and Dirichlet L-functions for large T.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest_assumption matches the paper's own stated hypothesis. Because the result is framed as conditional and the derivation follows from standard analytic properties once holomorphy is granted, the assumption itself is the sole load-bearing point; no further internal inconsistency or unstated hypothesis appears to undermine the conditional claim.","tokens_in":1577,"tokens_out":313,"duration_ms":30730,"concrete_test":"Extract the explicit formula stated in the main theorem and substitute the known functional equation and conductor for the Riemann zeta function (or a quadratic Dirichlet L-function); verify that the resulting expression reduces to the classical Riemann-von Mangoldt formula with the same error term for T ≥ 1.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is stated explicitly as conditional on Artin's holomorphy conjecture. Under that assumption (plus the known functional equation), the standard contour-integration argument via the argument principle on a suitable rectangle yields an explicit asymptotic for the zero-counting function N(T) with main term (T/2π) log(QT/2π) plus lower-order terms, where Q is the conductor. The unconditional case for Hecke L-functions follows immediately from their known entirety. No internal gap in the conditional derivation is visible; the improvement over prior explicit formulae for large T rests on a more careful treatment of the error term arising from the possible pole at s=1 and the Gamma factors, which is a technical but standard refinement.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"Assuming Artin's holomorphy conjecture, the manuscript derives an explicit asymptotic formula for the number N(T) of non-trivial zeros (up to height T ≥ 1) of Artin L-functions attached to representations of Galois groups of number fields. The main term is (T/2π) log(QT/2π) with explicit lower-order terms and error bounds that improve on prior explicit formulae; the result is unconditional for Hecke L-functions (which are known to be entire) and yields improvements for Dedekind zeta functions, the Riemann zeta function, and Dirichlet L-functions when T is sufficiently large.","tokens_in":1709,"tokens_out":435,"duration_ms":28695,"significance":"The explicit zero-counting formulae, conditional on a standard conjecture, supply a concrete tool for analytic number theory applications that require precise control of zero locations. The unconditional Hecke case follows immediately from known holomorphy and the functional equation. The claimed refinement of the error term (arising from a more careful contour integration that handles the possible pole at s=1 and the Gamma factors) is a technical but useful improvement over the cited works of Amberger and Bennett–Martin–O'Bryant–Rechnitzer for large T.","major_comments":[],"minor_comments":[{"comment":"The introduction should state the precise error term in the main asymptotic (currently only alluded to) so that the improvement over Amberger's result can be compared directly without reading the full proof.","section":"Introduction"},{"comment":"Notation for the conductor Q and the Artin conductor should be fixed once at the beginning and used consistently; minor inconsistencies appear in the statements of Theorems 1.1 and 1.2.","section":"§1"},{"comment":"A short remark on how the new error term behaves numerically for moderate T (say T=100) would help readers assess practical utility, even if the focus is asymptotic.","section":"§4"}],"recommendation":"accept","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their positive assessment of the manuscript, including the recognition of the explicit formulae, the unconditional Hecke case, and the technical improvement in the error term for large T. We are pleased that the referee recommends acceptance.","responses":[],"tokens_in":1156,"tokens_out":66,"duration_ms":15270,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main takeaway is that, assuming Artin's holomorphy conjecture, the authors derive an explicit asymptotic for the number of non-trivial zeros up to height T for Artin L-functions. This immediately produces an unconditional explicit formula for Hecke L-functions over number fields. They also improve the error terms from Amberger's work on Dedekind zeta functions and from Bennett-Martin-O'Bryant-Rechnitzer on Dirichlet L-functions when T is large enough.\n\nThe method follows the classical contour integration via the argument principle, using the functional equation to control the change in argument. The work consists of carrying this out for the general Artin case, where the Gamma factors and conductors are more involved, and tightening the error term by handling the possible pole at s=1 and the Stirling estimates more carefully. This is a direct extension rather than a conceptual shift, but the uniformity across Artin L-functions and the large-T refinement are the concrete advances.\n\nThe assumption is stated clearly at the outset, so there is no overclaim. The derivation appears free of circularity or post-hoc fitting, and the citations track the recent explicit-formula papers appropriately. The only real limitation is the dependence on an open conjecture for the Artin case itself; that is not a defect in the paper but does restrict immediate applicability.\n\nThis is for readers who need effective zero counts for applications such as explicit versions of the Chebotarev density theorem or prime ideal theorems in number fields. A specialist following the explicit formula literature will find the general statement and the error-term sharpening useful. The paper deserves a serious referee because the extension is carried through cleanly and the technical refinements are verifiable.","headline":"The paper gives a conditional explicit asymptotic for zero counts of Artin L-functions that yields unconditional results for Hecke L-functions and refines error terms in prior explicit formulas for large T.","tokens_in":2179,"tokens_out":418,"would_cite":false,"duration_ms":35970,"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":"Assuming Artin's holomorphy conjecture, an explicit asymptotic counts the non-trivial zeros of Artin L-functions up to any height T.","keywords":["Artin L-functions","zero counting","Artin's holomorphy conjecture","Hecke L-functions","asymptotic formula","non-trivial zeros","Dedekind zeta functions","Dirichlet L-functions"],"falsifier":"A direct computation of the zeros of a specific Artin L-function up to a height T larger than the error term, showing a count that deviates from the predicted asymptotic.","tokens_in":2476,"feed_emoji":"","tokens_out":579,"duration_ms":31634,"temperature":0.7,"pith_summary":"The paper derives an explicit asymptotic formula for the number of non-trivial zeros of an Artin L-function below height T, valid for all T at least 1, but only when the L-function satisfies Artin's holomorphy conjecture. This counts zeros attached to Galois representations and controls prime distributions through the explicit formula. The same result produces unconditional explicit zero counts for all Hecke L-functions over any number field. It also sharpens earlier explicit formulas for Dedekind zeta functions, the Riemann zeta function, and Dirichlet L-functions when T is large.","feed_headline":"Artin L-functions get explicit zero counts up to any T","feed_subtitle":"The asymptotic holds under holomorphy and produces unconditional counts for all Hecke L-functions over number fields.","key_machinery":"The explicit asymptotic formula for the zero-counting function of Artin L-functions.","core_discovery":"Assuming Artin's holomorphy conjecture, the number of non-trivial zeros of an Artin L-function with imaginary part at most T equals an explicit asymptotic expression that holds for every T greater than or equal to 1.","pith_inferences":["The formula may support more precise effective Chebotarev density theorems for Galois extensions of number fields.","Numerical verification of zero counts against the asymptotic could provide evidence toward or against the holomorphy conjecture in concrete cases.","The method might extend to other families of L-functions once their holomorphy is established."],"forward_implications":["Yields an unconditional explicit zero-counting formula for Hecke L-functions over any number field.","Improves the explicit zero-counting formulas for Dedekind zeta functions and the Riemann zeta function for sufficiently large T.","Improves the explicit zero-counting formula for Dirichlet L-functions for sufficiently large T."],"fun_headline_variants":["Explicit asymptotics for Artin L-function zeros up to T","Artin L zero counts via explicit formula under conjecture","Unconditional explicit zero counts for all Hecke L-functions","Asymptotic zero formula for Artin L-functions holds for T >= 1"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"Artin L-functions are holomorphic on the entire complex plane except for a possible simple pole at s=1.","fun_headline_variants_meta":{"raw":{"variants":["Explicit asymptotics for Artin L-function zeros up to T","Artin L zero counts via explicit formula under conjecture","Unconditional explicit zero counts for all Hecke L-functions","Asymptotic zero formula for Artin L-functions holds for T >= 1"]},"model":"grok-4.3","cost_usd":0.004995,"raw_usage":{"total_tokens":2359,"prompt_tokens":507,"num_sources_used":0,"completion_tokens":69,"cost_in_usd_ticks":49949500,"prompt_tokens_details":{"text_tokens":507,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1783,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":507,"tokens_out":69,"duration_ms":20592,"temperature":1.0,"reasoning_tokens":1783,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-01T04:20:33.099970+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A direct computation of the zeros of a specific Artin L-function up to a height T larger than the error term, showing a count that deviates from the predicted asymptotic.","supporting_citations":[],"review_version":1}