An analogue of a formula for Chebotarev Densities

Biao Wang

An analogue of a formula for Chebotarev Densities

T0 review · 0 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read For every integer m≥2, a λ_m-weighted sum over prime-divisor classes equals the Chebotarev density |C|/|G|.

desk verdict 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. read the letter →

arxiv 1908.05404 v3 pith:JKOTCLOY submitted 2019-08-15 math.NT

Biao Wang This is my paper · ORCID

classification math.NT MSC 11N1311R45

keywords ChebotarevdensitysmallestprimedivisorlargestLiouvillefunctionMöbiusdualitynumbertheoremDirichletseries

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

The 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.

The 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.

Who 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.

Referee Report

0 major / 6 minor

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.

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.

minor comments (6)

[Section 4, Eq. (4.10)] 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 2, Remark 2.2] 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 2, Eqs. (2.6) and (2.11)] 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 4, Eq. (4.5)] 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 4, Theorem 4.4] 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 3, Remark 3.2] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the proof derives the bridge identity from the Duality Lemma and uses external theorems, not the claim being proved.

full rationale

The derivation chain is self-contained in the relevant sense. The reader's flagged identity (4.5) is not an unproved assumption: setting g(n)=λ_m(n)f(p(n)), the Duality Lemma 3.1 states that ∑_{d|n}g(d)=-f(P_m(n)); Möbius inversion then gives g(n)=-∑_{d|n}μ(n/d)f(P_m(d)), which is exactly the first equality of (4.5). The subsequent estimates rely on the prime number theorem / Mertens bound, Ivić–Pomerance's estimate for large prime factors, and Dawsey's external formula for μ; all of these are genuine outside results, not the target theorem. The constant C_m is defined via an integral and its value is irrelevant to the limit, since it multiplies ∑_{n≤√x}μ(n)/n→0. There are no author self-citations, no imported uniqueness theorem, and no fitted parameter renamed as a prediction. The informal m→∞ remark is not used in the proof. The few typographical slips (e.g., 'P(n)' instead of 'P(d)' near (4.10)) are presentation errors, not circularity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 2 invented entities

The theorem is proven using standard analytic number theory. The only genuinely new mathematical objects are the functions λ_m and P_m, which are explicitly defined and shown to have the required properties. The proof imports three external results: Dawsey's Chebotarev density formula, the Ivić-Pomerance estimate for numbers with a large prime factor, and the prime number theorem (via the estimate (4.7)). No data fitting or free parameters are introduced.

assumptions (4)

domain assumption Dawsey's formula for Chebotarev densities (Theorem 1 of [4])
Used in the proof of Theorem 4.4 to evaluate the µ-partial sums that yield the main term |C|/|G|.
domain assumption Ivić-Pomerance estimate for sums over integers with a large prime factor (Theorem 4.1 from [8])
Used to prove Corollary 4.2, which bounds the difference between P_m and P.
standard math Prime number theorem (and the estimate ∑_{n≤x} μ(n)/n = O(exp(-c (log x)^{1/2})))
Used in Lemma 2.4 and in the proof of Theorem 4.4 to estimate the tail sums S2 and the μ partial sums.
standard math Standard Dirichlet series identities for ζ(s), 1/ζ(s), and ζ(ms)
Used to define λ_m and prove its multiplicative properties in Lemma 2.1.

invented entities (2)

λ_m(n), the coefficients of ζ(ms)/ζ(s) independent evidence
purpose: Generalizes the Liouville function and Möbius function; the central weight in the new density formula.
The function is explicitly defined via a Dirichlet series and has a closed form λ_m(n) = ∑_{d^m | n} μ(n/d^m), so its arithmetic meaning is fixed independently of any fitted constants.
P_m(n), the largest prime factor of n whose exponent is not divisible by m independent evidence
purpose: The dual prime factor function that appears in the Duality Lemma and the proof of Theorem 4.4.
The function is well-defined on integers and equals the usual largest prime factor P(n) when all exponents are coprime to m; no free parameters are involved.

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Cite this review

Pith. "Pith review of An analogue of a formula for Chebotarev Densities." pith.science (2026). https://pith.science/paper/JKOTCLOY

@misc{pith2026190805404,
  author       = {Pith},
  title        = {Pith review of: An analogue of a formula for Chebotarev Densities},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JKOTCLOY}},
  note         = {Machine review of arXiv:1908.05404}
}

abstract

In this short note, we show an analogue of Dawsey's formula on Chebotarev densities for finite Galois extensions of $\mathbb{Q}$ with respect to the Riemann zeta function $\zeta(ms)$ for any integer $m\geqslant2$. Her formula may be viewed as the limit version of ours as $m\to\infty$.

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Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

[1]

Alladi and P

K. Alladi and P. Erd¨ os, On an additive arithmetic functi on, Paciﬁc J. Math. 71(2)(1977) 275–294

work page 1977
[2]

Alladi, Duality between prime factors and an applicat ion to the prime number theorem for arithmetic progressions, J

K. Alladi, Duality between prime factors and an applicat ion to the prime number theorem for arithmetic progressions, J. Number Theory 9(4)(1977) 436–451

work page 1977
[3]

Broughan, Equivalents of the Riemann hypothesis I: Arithmetic equiva lents (Cam- bridge University Press, 2017)

K. Broughan, Equivalents of the Riemann hypothesis I: Arithmetic equiva lents (Cam- bridge University Press, 2017)

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Dawsey, A new formula for Chebotarev densities, Res

M.L. Dawsey, A new formula for Chebotarev densities, Res. Number Theory 3(2017) Article 27, 13 pp

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[5]

H. G. Diamond, Elementary methods in the study of the distribution of prime numbers, Bull. Amer. Math. Soc. (N.S.) 7(3)(1982) 553–589

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[6]

Ferenczi, J

S. Ferenczi, J. Ku/suppress laga-Przymus, and M. Lema´ nczyk, Sarnak’s conjecture: what’s new, in Ergodic theory and dynamical systems in their interactions with arithmetics and combinatorics, ed. S. Ferenczi, J. Ku/suppress laga-Przymus, and M. Lema´ nczyk, Lecture Notes in Math. Vol 2213 (Springer, 2018), pp. 163–235

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[8]

Ivi´ c and C

A. Ivi´ c and C. Pomerance, Estimates for certain sums involving the largest prime factor of an integer, in Topics in classical number theory, Vol. I, II (Budapest, 198 1), ed. G. Hal´ asz, Colloq. Math. Soc. J´ anos Bolyai Vol 34 (North-Holland, 1984), pp. 769–789

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Show all 9 references

[9]

Sweeting and K

N. Sweeting and K. Woo, Formulas for Chebotarev densitie s of Galois extensions of number ﬁelds, Res. Number Theory 5(1)(2019) Article 4, 13 pp

2019

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Alladi and P

K. Alladi and P. Erd¨ os, On an additive arithmetic functi on, Paciﬁc J. Math. 71(2)(1977) 275–294

work page 1977

[2] [2]

Alladi, Duality between prime factors and an applicat ion to the prime number theorem for arithmetic progressions, J

K. Alladi, Duality between prime factors and an applicat ion to the prime number theorem for arithmetic progressions, J. Number Theory 9(4)(1977) 436–451

work page 1977

[3] [3]

Broughan, Equivalents of the Riemann hypothesis I: Arithmetic equiva lents (Cam- bridge University Press, 2017)

K. Broughan, Equivalents of the Riemann hypothesis I: Arithmetic equiva lents (Cam- bridge University Press, 2017)

work page 2017

[4] [4]

Dawsey, A new formula for Chebotarev densities, Res

M.L. Dawsey, A new formula for Chebotarev densities, Res. Number Theory 3(2017) Article 27, 13 pp

work page 2017

[5] [5]

H. G. Diamond, Elementary methods in the study of the distribution of prime numbers, Bull. Amer. Math. Soc. (N.S.) 7(3)(1982) 553–589

work page 1982

[6] [6]

Ferenczi, J

S. Ferenczi, J. Ku/suppress laga-Przymus, and M. Lema´ nczyk, Sarnak’s conjecture: what’s new, in Ergodic theory and dynamical systems in their interactions with arithmetics and combinatorics, ed. S. Ferenczi, J. Ku/suppress laga-Przymus, and M. Lema´ nczyk, Lecture Notes in Math. Vol 2213 (Springer, 2018), pp. 163–235

work page 2018

[7] [7]

G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, sixth edition (Oxford University Press, 2008)

work page 2008

[8] [8]

Ivi´ c and C

A. Ivi´ c and C. Pomerance, Estimates for certain sums involving the largest prime factor of an integer, in Topics in classical number theory, Vol. I, II (Budapest, 198 1), ed. G. Hal´ asz, Colloq. Math. Soc. J´ anos Bolyai Vol 34 (North-Holland, 1984), pp. 769–789

work page 1984