{"id":"66b3e1f2-855e-4728-8ce7-a5a0f07dae04","arxiv_id":"2512.23302","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Under GRH alone, square-root-weighted prime counts in residue classes satisfy the Chebyshev-bias asymptotic -M(a,b) log log x + C on a set of x of natural density 1.","lead":"This paper proves that, assuming the Generalized Riemann Hypothesis, the known Chebyshev bias in prime distribution — more primes in non-square residue classes — holds with natural density one for a square-root-weighted prime count, without needing extra hypotheses on zero spacings. It supplies explicit error terms and an elementary proof, though the main asymptotic was independently obtained by Sheth (2025).","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the density-1 proof is sound under GRH; the only gap is a minor epsilon-shift in applying Lemma 2.2, not a threat to the claim.","rationale":"The reader's verdict is CONDITIONAL, and I agree that the proof should be read with care, but I do not find a load-bearing threat to the central claim. The threshold mismatch in Theorem 3.1 is real but easily fixed and does not change the conclusion. The moment-bound lemma is the true engine of the paper, and its internal estimates appear consistent; the claimed (Ck)^{4k} exponent is sufficient for the summability argument, whereas a weaker (Ck)^{8k} bound would break it. The GRH dependence is intrinsic to the conditional theorem and is not a flaw. Since the same asymptotic has independent support from Sheth's work, the central claim is credible. No verdict change is warranted.","tokens_in":10796,"tokens_out":53005,"duration_ms":440451,"concrete_test":"Re-derive the S2 zero-counting bound in the appendix and check whether the factor √(2k) log(2k(|ρ|+1)) can be absorbed as (Bk)^{4k-2}; then re-run the Markov estimate in Lemma 2.2 with threshold (log y)^{3+ε}/2 (replace ε by ε/2 on the RHS) and confirm the dyadic sum ∑ 2^{-kε/4} converges. If either step fails, Lemma 2.2's tail bound weakens and Theorem 1.1's density-1 conclusion would not follow.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No load-bearing concern identified. The central claim rests on Lemma 2.1's uniform moment bound (Ck)^{4k}; this exponent is exactly what makes the dyadic tail in Lemma 2.2 summable. I checked the appendix's S1/S2 split: S1 is bounded independently of Y, S2 is O(Y(Bk)^{4k-2}), and the symmetry step replacing signed sums by all-plus sums is legitimate because the zero multiset is closed under γ↦-γ and conjugation preserves |a_n|/|ρ_n|. The zero-counting estimate in S2 is unconditional and appropriate, since the condition forces |γ_{n_{2k}}|≈|s|. The only real defect in the written proof is in Theorem 3.1: Lemma 2.2 is stated for threshold (log y)^{3+ε}, but the proof applies it to (log y)^{3+ε}/2. This is absorbed by the Y^{o(1)} factor in the Markov bound (or by replacing ε with ε/2), so the dyadic estimate remains valid. This is a presentation issue, not a load-bearing flaw. GRH is genuinely the only hypothesis used; the proof does not secretly require LI.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the weighted prime counting function π_{1/2}(x;q,a) = ∑_{p≤x, p≡a mod q} p^{-1/2}. Under GRH for all non-principal Dirichlet characters modulo q, it proves that for distinct invertible residue classes a,b and every ε>0, the set of x≥2 satisfying |π_{1/2}(x;q,a)-π_{1/2}(x;q,b)+M(q;a,b) log log x - C| ≤ (log log x)^{3+ε}/log x has natural density 1. It also proves an analogous density-1 statement for partial Euler products and an asymptotic formula for the Cesàro mean of π_{1/2}. The method is to relate π_{1/2} to π by summation by parts, use the explicit formula, establish a uniform-in-k moment bound for the normalized error term Δ(y;t), and then apply Markov's inequality plus a finite-logarithmic-measure argument to convert almost-everywhere logarithmic control into a natural-density-one statement.","tokens_in":11095,"tokens_out":8757,"duration_ms":85232,"significance":"If the proof is correct, the paper removes the linear-independence hypothesis (LI) and the deep Riemann hypothesis (DRH) for this weighted formulation of Chebyshev's bias, obtaining a natural density-one statement under GRH alone, with explicit error terms rather than an implicit o(1). The key quantitative novelty is Lemma 2.1, a uniform moment bound of the form (Ck)^{4k}, which is exactly strong enough to make the dyadic tail summable. The proof uses no fitted parameters: the bias constant M(q;a,b) is built from the arithmetic quantities r(a) and m_χ and is part of the statement, not tuned to match the conclusion. The paper is also transparent about independent concurrent work by Sheth. The main risk lies in the density of the technical appendix, but the argument is coherent and the estimates are plausible; I found no circularity or hidden use of assumptions stronger than GRH.","major_comments":[],"minor_comments":[{"comment":"Lemma 2.2 is stated for the threshold (log y)^{3+ε}, but in the proof of Theorem 3.1 it is applied to the threshold (log y)^{3+ε}/2. This is harmless for large y, since (log y)^{3+ε}/2 = (log y)^{3+ε/2+o(1)} uniformly, but the mismatch should be acknowledged explicitly, e.g. by replacing ε with ε/2 or noting that the constant is absorbed.","section":"Section 3, proof of Theorem 3.1"},{"comment":"The decomposition into S1 and S2 is not literally a partition as written: S1 is defined using |γ_{n1}+···+γ_{nk}|, whereas S2 uses |γ_{n1}+···+γ_{n2k}|. For the subsequent estimates to make sense, S1 should be defined using the full 2k-sum, as the complementary case to S2. This appears to be a typographical error rather than a mathematical gap, but it should be corrected.","section":"Appendix, proof of Lemma 2.1"},{"comment":"The assertion that (log y)^{2k} ≥ Y for all y ≥ Y follows from the choice of k as the least even integer greater than log Y / log log Y, but this one-line justification is omitted. Adding it would improve readability.","section":"Lemma 2.2, proof"},{"comment":"The statement of Lemma 2.1 uses a constant C = C(t), while the proof introduces an auxiliary constant B depending only on q. It would be clearer to state explicitly that the final constant absorbs B, c_q, and the implied constants depending on q and t.","section":"Appendix, Lemma 2.1"},{"comment":"The note in the abstract about independent work by Sheth is unconventional; this information would better fit in the introduction or acknowledgments. It does not affect the mathematics, but it is worth adjusting for the published version.","section":"Abstract"}],"recommendation":"minor_revision","confidential_remarks":"I read the manuscript in good faith and found no load-bearing technical error. The only substantive issue is the ε-shift in the application of Lemma 2.2 in Theorem 3.1, which is absorbed by the Y^{o(1)} slack, and the S1/S2 definition typo in the appendix; both are easily fixable. The paper is a real contribution if the appendix survives scrutiny, and the explicit acknowledgment of Sheth's independent work is a positive feature. The editor may wish to verify that the relation to Sheth's paper is described accurately, but I do not see this as an obstacle."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Solid paper, worth a serious referee. It proves a density-one Chebyshev bias statement for the inverse-square-root weighted prime count under GRH alone, with explicit error terms. The main asymptotic was already implied by Sheth's independent work, and the paper says so plainly; the genuinely new pieces are the explicit error in Theorem 1.1, the density-one DRH version in Theorem 1.3, and the exact mean estimate in Theorem 1.4, plus a self-contained proof that does not rely on the stronger machinery.\n\nThe technical core holds up. Lemma 2.1 gives a uniform moment bound (Ck)^{4k}, and the appendix's refinement of Puchta is the crucial step; the dyadic tail in Lemma 2.2 sums because the exponent is right. The summation-by-parts derivation of (3.1) is clean, and the finite-logarithmic-measure argument converting to natural density is standard. I checked the zero-sum manipulations in the appendix, including the S1/S2 split and the symmetry step; they are legitimate. GRH is genuinely the only hypothesis used; there is no hidden LI.\n\nThe soft spots are minor. In Theorem 3.1, Lemma 2.2 is stated with threshold (log y)^{3+ε}, but the proof needs it at (log y)^{3+ε}/2. That's a presentation issue, not a load-bearing flaw; an epsilon shift or absorbing the factor into Y^{o(1)} fixes it. The appendix is dense and not machine-checked, but it is the kind of calculation that a competent referee can verify. The paper is conditional on GRH, so it does not settle the unweighted Chebyshev bias question, and the offset is a limitation of the whole approach rather than a flaw. The novelty is lower because of Sheth, but the explicit error terms and Theorem 1.4 are new, and the paper is honest about the overlap.\n\nFor an analytic number theory reading group this is a good candidate, especially if you want to see how to get natural density without LI. I would cite it if I work on weighted prime counting or Chebyshev bias. I would definitely send it to peer review; the main proof is sound and the minor epsilon issue is easily fixed.","headline":"A solid GRH-conditional result that removes LI/DRH for a weighted Chebyshev bias, with honest credit to Sheth and only a minor epsilon-shift gap.","tokens_in":11560,"tokens_out":1855,"would_cite":true,"duration_ms":17447,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11N13","11M26"],"pacs":[],"model":"deepseek-v4-flash","headline":"Under GRH alone, Chebyshev's bias for square-root-weighted prime counts is a density-one theorem.","keywords":["Chebyshev's bias","square-root weighted prime counts","Dirichlet L-functions","Generalized Riemann Hypothesis","density one","partial Euler products","moments of prime-counting error"],"falsifier":"The decisive check is to look for a nontrivial zero of a Dirichlet L-function modulo q off the critical line; even one such zero removes the cancellation in the exponential sums over γ that the moment bound requires. Short of that, one can numerically test Lemma 2.1 for q=4 and t=1_{3}−1_{1} on long dyadic intervals: the lemma asserts (1/Y)∫_{log2}^{Y}|Δ(y;t)|^{2k}dy ≤ (Ck)^{4k} uniformly in k, so a run over large Y and several k that exceeds this bound would refute the load-bearing moment estimate and hence the proof of Theorem 1.1.","tokens_in":10683,"feed_emoji":"🔢","tokens_out":5623,"duration_ms":55148,"temperature":0.7,"pith_summary":"Chebyshev noticed that primes are usually more numerous in non-square residue classes, but proving this bias rigorously has required extra assumptions about the zeros of L-functions. This paper shows that if each prime is weighted by its inverse square root, the bias becomes a density-one statement under the Generalized Riemann Hypothesis alone—no linear independence hypothesis and no stronger deep Riemann hypothesis. The main theorem gives an explicit error window: away from a set of natural density zero, the weighted difference π1/2(x;q,a)−π1/2(x;q,b) equals the predicted bias −M(q;a,b) log log x + C up to error (log log x)^(3+ε)/log x. The argument uses the standard explicit formula, summation by parts, and a new uniform bound on high moments of the normalized prime-count error. If correct, Chebyshev's bias for these weighted counts is a theorem, not a heuristic, under GRH.","feed_headline":"GRH alone proves Chebyshev bias for weighted primes","feed_subtitle":"Weighting primes by 1/√p turns the bias into a density-one theorem, no extra zero assumptions needed.","key_machinery":"The key object is the normalized error Δ(y;t) = y π(e^y;t)/e^(y/2) + 2M(t), an almost-periodic oscillatory sum over the zeros of Dirichlet L-functions. The proof hinges on a uniform high-moment estimate: for every k≥1, (1/Y)∫_{log2}^{Y}|Δ(y;t)|^{2k}dy ≤ (Ck)^{4k}, obtained by refining standard multisum estimates over zeros. Markov's inequality then bounds the large deviations of Δ by a set of finite logarithmic measure, which the paper converts into zero natural density. Summation by parts transfers these bounds from Δ to the weighted prime count π1/2, and the main bias term emerges from the integral term in that summation.","core_discovery":"The central claim is that GRH for all non-principal Dirichlet characters modulo q is sufficient for Chebyshev's bias in the square-root weighted counting function. For distinct invertible residue classes a and b, and every ε>0, the natural density of the set of x for which |π1/2(x;q,a)−π1/2(x;q,b)+M(q;a,b) log log x − C| ≤ (log log x)^(3+ε)/log x exists and equals 1. The constant M(q;a,b) encodes the bias: when no character has a zero at s=1/2, it reduces to (r(a)−r(b))/(2φ(q)), so the class containing more square roots among reduced residues has fewer weighted primes on a set of density one. The paper also proves a density-one statement for the partial Euler products that appear in the deep","pith_inferences":["Because the proof avoids linear independence, it suggests that other density-one bias phenomena—such as in Chebotarev or automorphic settings—might be reachable under GRH alone once suitable uniform moment bounds are available.","The uniform moment bound (Ck)^{4k} may be reusable in neighboring problems where large-deviation control of zero sums is needed, such as discrepancies of primes in short intervals or other weighted counting functions.","The error window (log log x)^(3+ε)/log x is likely far from optimal; a natural extension would be to shrink the exponent or determine the true limiting distribution of the normalized bias, which this method does not address.","The density-one statement identifies typical x but gives no explicit description of the exceptional set, so making the result quantitative for a specific modulus would require additional arithmetic work to compute the constant C explicitly."],"forward_implications":["For q=4, assuming GRH for the mod-4 character, the set where π1/2(x;4,3)>π1/2(x;4,1) has natural density 1, confirming Chebyshev's 1853 observation in weighted form.","The partial Euler products (log x)^(mχ) ∏_{p≤x}(1−χ(p)/√p)^(−1) converge to a nonzero constant ℓχ on a set of natural density 1, so the deep Riemann hypothesis conclusion holds for almost all x under GRH.","The mean value (1/x)∫_2^x (π1/2(u;q,a)−π1/2(u;q,b))du has the exact asymptotic −M(q;a,b) log log x + C + O(log log x/log x), matching what the deep Riemann hypothesis would predict.","If no relevant L-function vanishes at s=1/2, the bias constant is (r(a)−r(b))/(2φ(q)), so the reduced residue class with fewer square roots is the class with more weighted primes on a density-one set.","The method works for general functions t on reduced residues with mean zero against the principal character, and the author states the arguments extend to global fields."],"fun_headline_variants":["Weighted primes reveal Chebyshev bias under GRH alone","GRH suffices for Chebyshev bias in square-root weighted count","Density-one Chebyshev bias without zero-linear independence","Chebyshev's bias proven with only GRH, no extra zero assumptions","Square-root prime weights turn Chebyshev bias into density-one"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is GRH for all non-principal Dirichlet characters modulo q: every nontrivial zero of those L-functions must lie on the critical line Re(s)=1/2. If a single relevant zero sits elsewhere, the exponential sums over zeros no longer cancel with the regularity the proof needs.","fun_headline_variants_meta":{"raw":{"variants":["Weighted primes reveal Chebyshev bias under GRH alone","GRH suffices for Chebyshev bias in square-root weighted count","Density-one Chebyshev bias without zero-linear independence","Chebyshev's bias proven with only GRH, no extra zero assumptions","Square-root prime weights turn Chebyshev bias into density-one"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000261,"raw_usage":{"total_tokens":1446,"prompt_tokens":775,"completion_tokens":671,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":519,"completion_tokens_details":{"reasoning_tokens":581}},"tokens_in":519,"tokens_out":671,"duration_ms":6118,"temperature":1.0,"reasoning_tokens":581,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T13:40:40.083546+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"The decisive check is to look for a nontrivial zero of a Dirichlet L-function modulo q off the critical line; even one such zero removes the cancellation in the exponential sums over γ that the moment bound requires. Short of that, one can numerically test Lemma 2.1 for q=4 and t=1_{3}−1_{1} on long dyadic intervals: the lemma asserts (1/Y)∫_{log2}^{Y}|Δ(y;t)|^{2k}dy ≤ (Ck)^{4k} uniformly in k, so a run over large Y and several k that exceeds this bound would refute the load-bearing moment estimate and hence the proof of Theorem 1.1.","supporting_citations":[],"review_version":1}