{"id":"3b9e7799-527b-4e1d-a9fc-dea31c43a74e","arxiv_id":"1908.05404","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For each integer m at least 2, the weighted sum of the generalized Liouville function λ_m over integers whose smallest prime factor has a given Frobenius conjugacy class equals the expected density |C|/|G|.","lead":"This paper proves a family of new Chebotarev density formulas for finite Galois extensions of the rational numbers, one for each integer m, with the classical version of Dawsey appearing as the formal limit. The result matters to number theorists because it shows the same duality between smallest and largest prime factors extends to a wider class of arithmetic functions.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: identity (4.5) is valid and follows directly from Lemma 3.1 by Möbius inversion.","rationale":"The stress-test pass found the central argument sound. The only substantive concern in the reader's verdict was the unproved identity (4.5), but that identity is a direct two-line consequence of Lemma 3.1 and Möbius inversion, and the paper explicitly cites exactly those ingredients. I verified Lemma 3.1's j0 argument and the partial summation in Corollary 4.2; both are correct. The error bounds in Theorem 4.4 follow from Dawsey's theorem, the PNT estimate (4.7), and the tail bound (4.3); the notational reuse of C_m in (4.10) does not affect the argument since only boundedness of the constant is used. The remaining issues are typos (e.g., λ_m(x) for λ_m(n) in Remark 2.2) and an informal limit remark, which do not threaten Theorem 1.1. The verdict CONDITIONAL is acceptable as a request for clearer derivation, but no load-bearing correction is needed, hence UNCHANGED.","tokens_in":6666,"tokens_out":18356,"duration_ms":166942,"concrete_test":"For m=2 and m=3, with f(p)=1 if p≡1 mod 3 and f(1)=0, compute for all n≤10^5 both sides of the identity λ_m(n)f(p(n))=−∑_{d|n}μ(n/d)f(P_m(d)) and both sides of Lemma 3.1; a single mismatch would indicate a hidden exception, while agreement confirms the bridge on which Theorem 4.4 relies.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No significant objection identified. The reader's flagged identity (4.5) is not a hidden assumption. Setting g(n)=λ_m(n)f(p(n)), the Duality Lemma 3.1 states ∑_{d|n}g(d)=−f(P_m(n)); Möbius inversion yields g(n)=−∑_{d|n}μ(n/d)f(P_m(d)), which is exactly the first equality of (4.5). Lemma 3.1's proof is correct: the j0 argument isolates the largest prime whose exponent is not divisible by m, and all other terms vanish because a(p^α)−1=0 or a(d_{j+1})=0. The subsequent splitting at x^{1/2} and the bounds from Corollary 4.2 and (4.7) are standard; the reuse of C_m in (4.10) for the f-weighted constant is harmless because any constant multiple of ∑_{n≤√x}μ(n)/n is O(exp(−c√log x)). Remaining issues are typographical and presentational, not load-bearing.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces λ_m(n), the coefficients of ζ(ms)/ζ(s), a generalization of the Liouville function, and proves Theorem 1.1: for every finite Galois extension K/Q with Galois group G, the series -∑_{n≥2, [K/Q/p(n)]=C} λ_m(n)/n converges to |C|/|G| for each conjugacy class C. The proof proceeds via a duality lemma relating λ_m to the largest prime factor of n whose exponent is not divisible by m, a Möbius-inversion step that rewrites the λ_m-sum in terms of the corresponding μ-sum plus an error term, and estimates from Ivić-Pomerance to control the difference between the modified largest-prime-factor function P_m and the usual P. The paper also proves that the prime number theorem is equivalent to ∑_{n≥1} λ_m(n)/n = 0.","tokens_in":6890,"tokens_out":30171,"duration_ms":250851,"significance":"Assuming correctness, the result gives a new family of exact density formulas for Chebotarev densities, interpolating between the Möbius function (as m→∞) and the Liouville function (m=2). The proof is transparent and builds on Dawsey's formula, the Ivić-Pomerance estimates, and the prime number theorem. The paper includes a self-contained proof of the prime-number-theorem equivalence for λ_m (Lemma 2.4) and a clear duality lemma (Lemma 3.1). The key identity (4.5) is correct: it follows from Lemma 3.1 by convolving with μ and using ∑_{r|k} μ(r)=ε(k). The main theorem is a natural and likely publishable generalization of Dawsey's result, and the paper is carefully structured for a short note.","major_comments":[],"minor_comments":[{"comment":"The argument of f on the right side should be P(d) rather than P(n); as written, f(P(n)) is constant in d, and the displayed constant C_m does not follow from Corollary 4.2. The proof is unaffected because the weighted sum is bounded and its precise limit is multiplied by o(1), but the equation should be corrected or phrased with an unspecified bounded constant.","section":"Section 4, Eq. (4.10)"},{"comment":"The displayed sum should be ∑_{n≤x} λ_m(n), not λ_m(x); as written the formula is a typo and does not express the stated estimate.","section":"Section 2, Remark 2.2"},{"comment":"The notation is difficult to read where d^m is rendered as \"dm\" without superscripts; please clarify that the sums are over d^m and properly typeset the split at x^{1/2}.","section":"Section 2, Eqs. (2.6) and (2.11)"},{"comment":"Although the identity is correct, the derivation is highly compressed; adding the intermediate step (convolving Lemma 3.1 with μ and using ∑_{r|k} μ(r)=ε(k)) would substantially improve readability.","section":"Section 4, Eq. (4.5)"},{"comment":"The stated error term is exp(-c (log x)^{1/3}), while the estimates in (4.9) and (4.11) yield exp(-c (log x)^{1/2}); please clarify whether the cited [4, (10)] indeed gives the weaker exponent or whether the theorem can state the stronger error uniformly.","section":"Section 4, Theorem 4.4"},{"comment":"This dual identity is not used in the paper and its statement is hard to parse; consider removing it or expanding the explanation, since the notation d_j(n) is not fully defined in a self-contained way.","section":"Section 3, Remark 3.2"}],"recommendation":"minor_revision","confidential_remarks":"The paper is a short, carefully written note. I found no fatal mathematical errors; in particular, I verified the central identity (4.5) by hand, and the main theorem follows from the stated estimates and Dawsey's formula. The remaining issues are typographical and presentational, so a minor revision should suffice. The manuscript fits the scope of a number theory journal and is likely to be of interest to specialists in Chebotarev densities and arithmetic functions."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper does what it says: it proves a one-parameter family of Chebotarev density formulas, with Dawsey's µ-formula as the formal m→∞ limit. The main theorem (1.8) is new for fixed m≥2, and the setup—λ_m(n) as coefficients of ζ(ms)/ζ(s), plus the dual prime factor function P_m(n)—is a genuine adaptation, not just a cosmetic tweak. If you care about Alladi's duality or Chebotarev density formulas, this is worth your time.\n\nThe mathematics checks out. Lemma 2.1 gives the key properties of λ_m, and Lemma 2.4 proves the PNT equivalence for λ_m in the expected way. The Duality Lemma 3.1 is correct: the j_0 argument isolates the largest prime whose exponent is not divisible by m, and the earlier terms vanish because a(p^α)−1=0 or a(d_{j+1})=0. The proof of Theorem 4.4 is a clean reduction to Dawsey's formula plus error estimates from the Ivić–Pomerance bound. The reader's weakest-assumption concern is not actually a problem: identity (4.5) follows directly from Lemma 3.1 by Möbius inversion with g(n)=λ_m(n)f(p(n)). The paper does not spell that derivation out, and it should—citing \"as [2, (2.35)]\" is too terse—but there is no hidden assumption.\n\nThe soft spots are mostly presentational. There are typos and OCR-unfriendly notation throughout (d^m rendered as \"dm\", a garbled sum in Lemma 2.4, the function P_m defined by a mangled character). The reuse of C_m in (4.10) for a different weighted constant is sloppy but harmless. The limit remark (Remark 1.2) is informal, but it is not used in the proof. None of these affect correctness. The citation pattern is appropriate: Alladi, Dawsey, Sweeting–Woo, Ivić–Pomerance, and the standard PNT references are all there, and the new work builds on them without circularity.\n\nWho is this for? Anyone working on Chebotarev density formulas, Alladi's duality, or the λ_m generalization of the Liouville function. It is a short note with a real theorem inside. Send it to a serious referee; it deserves refereeing rather than a desk reject. With the derivation of (4.5) made explicit and the typos cleaned up, this should be acceptable. I would take the refereeing if asked.","headline":"A short, correct note that genuinely generalizes Dawsey's Chebotarev density formula to the family λ_m(n) from ζ(ms)/ζ(s); the main theorem is new, the proof is sound, and the one flagged 'gap' in identity (4.5) is actually a minor presentation issue, not a load-bearing flaw.","tokens_in":7433,"tokens_out":2263,"would_cite":true,"duration_ms":23352,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11N13","11R45"],"pacs":[],"model":"deepseek-v4-flash","headline":"For every integer m≥2, a λ_m-weighted sum over prime-divisor classes equals the Chebotarev density |C|/|G|.","keywords":["Chebotarev density","smallest prime divisor","largest prime divisor","Liouville function","Möbius function","duality","prime number theorem","Dirichlet series"],"falsifier":"Compute both sides of (4.5) directly for a finite Galois extension such as K=Q(ζ_3) and a range of n with p(n) in a fixed class, with f equal to the indicator of that class; any mismatch would falsify the proof. A numerical evaluation of the partial sums in Theorem 4.4 for m=2 should tend to |C|/|G| at the stated rate; a clear deviation would falsify the theorem.","tokens_in":6454,"feed_emoji":"🔢","tokens_out":6122,"duration_ms":58974,"temperature":0.7,"pith_summary":"This paper proves an exact analogue of the known Chebotarev-density formula in which the Möbius function µ(n) is replaced by the coefficient λ_m(n) of ζ(ms)/ζ(s), for any fixed integer m≥2. For every finite Galois extension K of Q and every conjugacy class C of its Galois group, the claim is that −∑_{n≥2, [K/Q/p(n)]=C} λ_m(n)/n = |C|/|G|. Since λ_2 is the Liouville function and λ_m converges coefficientwise to µ as m→∞, the older formulas for µ become the limiting case of this family. The proof works by comparing the λ_m-weighted partial sums with the µ-weighted ones, using a duality identity and an estimate that keeps a modified largest-prime-divisor function close to the usual one.","feed_headline":"Each m≥2 gives an exact Chebotarev density identity","feed_subtitle":"Weighting by coefficients of ζ(ms)/ζ(s), the smallest-prime-divisor sum over each class is |C|/|G|.","key_machinery":"The load-bearing object is the Duality Lemma: for any arithmetic function f with f(1)=0, ∑_{d|n} λ_m(d) f(p(d)) = −f(P_m(n)), where p(d) is the smallest prime divisor and P_m(n) is the largest prime factor of n whose exponent is not divisible by m (with P_m(n)=1 when n is a perfect m-th power). This identity converts the λ_m-weighted sum over smallest prime divisors into a sum over the modified largest prime factor, allowing the author to compare with the known µ formula. A second ingredient is the estimate that P_m(n) differs from the ordinary largest prime factor P(n) only on a sparse set, so the difference of the two weighted sums is controlled by the known large-prime-factor asymptotics.","core_discovery":"The central discovery is a one-parameter family of exact density identities. For each integer m≥2, the Dirichlet-series ratio ζ(ms)/ζ(s) defines a multiplicative function λ_m, and the paper proves that the smallest-prime-divisor sum weighted by λ_m over any conjugacy class of the Galois group of a finite Galois extension of Q converges to the Chebotarev density of that class. This is a genuine extension of the µ-based formula: the m=2 case is a Liouville-function analogue, and the limit m→∞ returns the µ formula because ζ(ms)→1 and λ_m(n)→µ(n). The theorem is stated with an explicit saving in the error term, of order exp(−c(log x)^{1/3}) in the partial sums.","pith_inferences":["We infer that the same coefficient method should work for ratios ζ(as)/ζ(s) or products of zeta factors, producing further exact density identities; the paper leaves this open.","We infer that the explicit error term exp(−c(log x)^{1/3}) should be testable numerically for small m and small Galois extensions, and its sharpness has not been studied.","We infer that the duality identity probably extends to λ_m with other arithmetic functions f beyond the Artin-symbol indicator, which would yield weighted analogues in other sieving contexts.","We infer that the author's remark about number fields could be realized by replacing primes of Q with prime ideals, giving Chebotarev-density formulas for λ_m over global fields."],"forward_implications":["For each m≥2, formula (1.8) provides an exact identity: the λ_m-weighted density of primes whose Artin symbol lies in C is |C|/|G|.","The m=2 case yields a Liouville-function analogue of the Chebotarev density formula, so the result applies when λ_m(n)=(−1)^{Ω(n)}.","Letting m→∞ recovers the µ formula, so the new identities include the older theorem as a limiting case and explain its structure as a family.","The cyclotomic specialization gives a congruence identity: for (ℓ,k)=1, the λ_m-weighted sum over n with p(n)=ℓ mod k is 1/φ(k).","Lemma 2.4 adds a new PNT equivalence: the prime number theorem holds precisely when ∑ λ_m(n)/n converges to 0 for a fixed m."],"supporting_citations":[{"why":"Supplies the duality method and the Möbius-inversion step (2.35) used to transform the λ_m sum.","marker":"[2]"},{"why":"Supplies the µ-based Chebotarev density formula and the error bound (10) that completes the comparison.","marker":"[4]"},{"why":"Supplies the asymptotic estimate for the largest prime divisor that makes P_m close to P and controls the difference terms.","marker":"[8]"}],"fun_headline_variants":["m≥2: new exact Chebotarev density formulas","Weighted smallest-prime sums recover Chebotarev densities","One parameter family of exact Chebotarev identities","Liouville to Möbius: Chebotarev density limit","Exact Chebotarev densities for every integer m≥2"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof's bridge is the unproved identity (4.5), which rewrites the λ_m-weighted smallest-prime-divisor sum as a double sum over µ and the modified largest-prime-divisor function; if that pointwise identity has exceptions or hidden conditions, the reduction to the known µ formula collapses.","fun_headline_variants_meta":{"raw":{"variants":["m≥2: new exact Chebotarev density formulas","Weighted smallest-prime sums recover Chebotarev densities","One parameter family of exact Chebotarev identities","Liouville to Möbius: Chebotarev density limit","Exact Chebotarev densities for every integer m≥2"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000868,"raw_usage":{"total_tokens":3662,"prompt_tokens":750,"completion_tokens":2912,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":366,"completion_tokens_details":{"reasoning_tokens":2825}},"tokens_in":366,"tokens_out":2912,"duration_ms":21707,"temperature":1.0,"reasoning_tokens":2825,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:15:30.979766+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute both sides of (4.5) directly for a finite Galois extension such as K=Q(ζ_3) and a range of n with p(n) in a fixed class, with f equal to the indicator of that class; any mismatch would falsify the proof. A numerical evaluation of the partial sums in Theorem 4.4 for m=2 should tend to |C|/|G| at the stated rate; a clear deviation would falsify the theorem.","supporting_citations":[{"cited_title":"Alladi, Duality between prime factors and an applicat ion to the prime number theorem for arithmetic progressions, J","cited_arxiv_id":null,"evidence_quote":"Supplies the duality method and the Möbius-inversion step (2.35) used to transform the λ_m sum."},{"cited_title":"Dawsey, A new formula for Chebotarev densities, Res","cited_arxiv_id":null,"evidence_quote":"Supplies the µ-based Chebotarev density formula and the error bound (10) that completes the comparison."},{"cited_title":"Ivi´ c and C","cited_arxiv_id":null,"evidence_quote":"Supplies the asymptotic estimate for the largest prime divisor that makes P_m close to P and controls the difference terms."}],"review_version":1}