Chebyshev's bias without linear independence

Mounir Hayani

Chebyshev's bias without linear independence

T0 review · 0 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read Under GRH alone, Chebyshev's bias for square-root-weighted prime counts is a density-one theorem.

desk verdict 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. read the letter →

arxiv 2512.23302 v3 pith:YOWUW4R6 submitted 2025-12-29 math.NT

Mounir Hayani This is my paper · ORCID

classification math.NT MSC 11N1311M26

keywords Chebyshev'sbiassquare-rootweightedprimecountsDirichletL-functionsGeneralizedRiemannHypothesisdensityonepartialEulerproductsmomentsofprime-countingerror

topics The Riemann Hypothesis

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

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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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

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.

The 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.

The 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.

For 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.

Referee Report

0 major / 5 minor

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.

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.

minor comments (5)

[Section 3, proof of Theorem 3.1] 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.
[Appendix, proof of Lemma 2.1] 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.
[Lemma 2.2, proof] 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.
[Appendix, Lemma 2.1] 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.
[Abstract] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central density-1 theorem is derived from GRH, the explicit formula, and independent uniform moment estimates; the bias constant M(q;a,b) is an arithmetic input, not a fitted parameter.

full rationale

The derivation chain is self-contained and does not reduce to its own inputs. The bias constant M(t) is defined in (2.2) from arithmetic data (r and m_chi), and the explicit formula (2.3) is an external standard identity; the proof then shows that pi_{1/2}(x;t)+M(t)log log x-C is small on a density-1 set. No parameter is fitted to the theorem's conclusion: M(t) is not tuned to match the result, and GRH is an external hypothesis used through explicit formulas and zero-counting bounds. Lemma 2.1's uniform moment bound is an independent technical contribution, and Lemma 2.2 legitimately converts it into a distributional statement. The proof does not rely on any self-citation: the paper cites no work of its own author, and the cited external results (Rubinstein-Sarnak, Puchta, Aoki-Koyama, Montgomery-Vaughan, etc.) are used as standard tools, not as unverified uniqueness theorems. The note about Sheth's independent work actually corroborates the result externally. The only minor issue in the written proof is an epsilon-shift when applying Lemma 2.2 in Theorem 3.1, but this is absorbed by choosing a slightly smaller epsilon and is a presentation issue, not a circular dependency.

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

The paper introduces no fitted parameters or invented entities. The constants C, L, ℓχ, and K are existential constants determined by the arithmetic input, not free parameters tuned to data. All non-standard assumptions are external (GRH) or standard analytic number theory background.

assumptions (4)

domain assumption GRH for all non-principal Dirichlet L-functions modulo q.
Stated in Theorems 1.1, 1.3, and 1.4; used in the explicit formula (2.1) and in the zero-moment estimates of Lemma 2.1.
standard math Standard explicit formula for ψ(x;χ) from Montgomery–Vaughan Theorem 12.12.
Invoked at equation (2.1) to expand ψ(x;t) in terms of zeros of Dirichlet L-functions.
standard math Uniform zero-counting bounds such as ∑_{|γ|≤T} 1/|ρ| ≪ (log T)^2 under GRH.
Used in the appendix to estimate the multisums over zeros in Lemma 2.1.
standard math Mertens-type estimates for ∑_{p≤x} χ(p^2)/p.
Used in the proof of Theorem 1.3 to derive the asymptotic for log F(x).

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Cite this review

Pith. "Pith review of Chebyshev's bias without linear independence." pith.science (2026). https://pith.science/paper/YOWUW4R6

@misc{pith2026251223302,
  author       = {Pith},
  title        = {Pith review of: Chebyshev's bias without linear independence},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YOWUW4R6}},
  note         = {Machine review of arXiv:2512.23302}
}

abstract

We confirm Chebyshev's observation that primes are strikingly more abundant in non-square residue classes modulo a fixed integer under the Generalized Riemann Hypothesis (GRH) by proving a (natural) density $1$ statement for prime counting functions in residue classes where each prime is weighted by its inverse square root. In contrast to the majority of the existing literature on the subject, we do not need to restrict to logarithmic densities to measure Chebyshev's bias, and we do not rely on any hypothesis on the zeros of $L$-functions that is stronger than GRH. Note: The same type of results presented here were independently proved by Arshay Sheth (2025) in the general context of automorphic forms, which implies our main asymptotic. While the spirit of the proofs is similar, Sheth develops an explicit formula for the partial Euler product and uses a result due to Gallagher (1980) to prove that some estimates hold outside a set of finite logarithmic measure. We share this independent work because it provides a completely self-contained and elementary proof relying only on the usual explicit formula, and it yields explicit error terms rather than an implicit $o(1)$ asymptotic.

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Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [1]

Aoki and S.-y

M. Aoki and S.-y. Koyama, Chebyshev’s bias against splitting and principal primes in global ﬁelds , J. Number Theory 245 (2023), 233–262, DOI 10.1016/j.jnt.2022.10.005. MR4517481

doi:10.1016/j.jnt.2022.10.005 2023

[2] [2]

de la Bret` eche and D

R. de la Bret` eche and D. Fiorilli, Moments of moments of primes in arithmetic progressions , Proc. Lond. Math. Soc. (3) 127 (2023), no. 1, 165–220, DOI 10.1112/plms.12542. MR4611407

doi:10.1112/plms.12542 2023

[3] [3]

de la Bret` eche, D

R. de la Bret` eche, D. Fiorilli, and F. Jouve, Moments in the Chebotarev density theorem: general class functions , Algebra Number Theory 19 (2025), no. 3, 481–520, DOI 10.2140/ant.2025.19.481. MR4879359

work page doi:10.2140/ant.2025.19.481 2025

[4] [4]

Chebyshev, Lettre de M

P. Chebyshev, Lettre de M. le Professeur Tch´ ebychev ` a M. Fuss sur un nouve au th´ eor` eme relatif aux nombres premiers contenus dans les formes 4n + 1 et 4n + 3 , Bull. Classe Phys. Acad. Imp. Sci. St. Petersburg (1853), 208

[5] [5]

Conrad, Partial Euler products on the critical line , Canad

K. Conrad, Partial Euler products on the critical line , Canad. J. Math. 57 (2005), no. 2, 267–297, DOI 10.4153/CJM-2005-012-6. MR2124918

work page doi:10.4153/cjm-2005-012-6 2005

[6] [6]

Devin, Chebyshev’s bias for analytic L-functions, Math

L. Devin, Chebyshev’s bias for analytic L-functions, Math. Proc. Cambridge Philos. Soc. 169 (2020), no. 1, 103–140

2020

[7] [7]

Fiorilli and F

D. Fiorilli and F. Jouve, Distribution of Frobenius elements in families of Galois ex tensions, J. Inst. Math. Jussieu 23 (2024), no. 3, 1169–1258

2024

[8] [8]

Hooley, On the Barban-Davenport-Halberstam theorem

C. Hooley, On the Barban-Davenport-Halberstam theorem. XVII , Proceedings of the Session in Analytic Number Theory and Diophantine Equations, Bonner Math. Schrifte n, vol. 360, Univ. Bonn, Bonn, 2003, pp. 5. MR2075630

2003