The asymptotic estimation of prime ideals in imaginary quadratic fields and Chebyshev's bias

Chenhao Tang; Chen Lin; Xuejun Guo

REVIEW 2 major objections 6 minor 11 references

The asymptotic estimation of prime ideals in imaginary quadratic fields and Chebyshev's bias

T0 review · 2 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper proves an explicit asymptotic for prime ideals in imaginary quadratic fields that satisfy both a norm congruence and an argument condition, with leading coefficient either zero or a field extension degree.

desk verdict A real extension of Coleman's theorem with a correct but under-verified key step in Section 2.7; worth serious refereeing. read the letter →

arxiv 2412.14792 v1 pith:GCM2OCG6 submitted 2024-12-19 math.NT

Chen Lin , Chenhao Tang , Xuejun Guo This is my paper · ORCID

classification math.NT MSC 11N0511K70

keywords asymptoticestimationquadraticformsChebyshev'sbiasimaginaryfieldsprimeidealsargumentequidistributionHeckeL-functionsChebotarevdensity

topics The Twin Prime Conjecture

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 that the angular equidistribution theorem for prime ideals in imaginary quadratic fields can be refined to count prime ideals whose norm satisfies a fixed congruence modulo $M$ at the same time as the argument lies in a prescribed interval. The main theorem gives a main term $$A(m,M,C,\mathfrak{a})\frac{(\varphi_2-\varphi_1)y}{2\pi h_{\mathfrak{a}}\varphi(M)\log x}\left(1+O(\tfrac{1}{\log x})\right)$$ for the number of such prime ideals, valid when the interval length exceeds $x^{-5/24+\epsilon}$ and $y>x^{19/24+\epsilon}$. The coefficient $A$ is a finite character sum that the paper evaluates via class field theory as either $0$ or the degree of a certain subfield of $\mathbb{Q}(\zeta_M)$. The payoff is that results for primes represented by binary quadratic forms, previously known only over all representable primes, now hold for primes in a fixed residue class modulo $M$, and the numerical data for $x^2+y^2$ and $x^2+xy+y^2$ exhibit repeated sign changes between residue classes, which the authors formulate as Chebyshev-bias conjectures.

What carries the argument

The engine is a character-weighted Euler product $L(\phi,\psi,s)=\prod_{\mathfrak p}(1-\phi(N(\mathfrak p))\psi(\mathfrak p)N(\mathfrak p)^{-s})^{-1}$, where $\phi$ runs over Dirichlet characters modulo $M$ and $\psi$ over characters of the ray class group $Cl(K,\mathfrak{a})$. Up to finitely many Euler factors this is a Hecke $L$-function; the pole condition at $s=1$ selects exactly the pairs with $\phi(N(\mathfrak p))\psi(\mathfrak p)=1$ for almost all $\mathfrak p$, which is the condition defining the coefficient $A$. A parallel route uses class field theory and the Chebotarev density theorem: the class condition and the norm congruence become one Frobenius element in $\mathrm{Gal}(L_3/K)$, and orthogonality of characters gives $A=[f_{L_2}:\mathbb Q]$ or $A=0$ according as the two Frobenius restrictions agree. To get the uniform error term, $L_3$ is placed inside a ray class field $K(\mathfrak{b})$, so the restricted primes are exactly $s=[K(\mathfrak{b}):L_3]$ ideal classes modulo $\mathfrak{b}$; applying the angular equidistribution theorem to those classes yields the asymptotic. The argument cutoff is carried by the unit-root character $\lambda(\mathfrak{a})=(\xi_{\mathfrak{a}}/|\xi_{\mathfrak{a}}|)^g$.

What would settle it

Take $K=\mathbb{Q}(i)$, $\mathfrak{a}=(1)$, the trivial ray class, $M=8$, $m=1$, and enumerate Gaussian primes with norm $p\equiv1\pmod8$ and argument in a fixed interval up to $x=10^7$. The theorem predicts a count asymptotic to $\frac{(\varphi_2-\varphi_1)y}{8\pi\log x}$ with error $O(1/\log x)$; if the count disagrees by more than that size, or if changing the auxiliary modulus $\mathfrak{b}$ in Section 2.7 changes the implied constant, the angular-compatibility step fails. A characteristic check is whether the same $A$ is obtained from the character sum and from the class-field formula in a case where $L_2$ is nontrivial.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem 1.2: for an imaginary quadratic field $K$, a ray class $C$ modulo $\mathfrak{a}$, and coprime integers $m,M$, the count of prime ideals $\mathfrak{p}\in C$ with $x-y\le N(\mathfrak{p})\le x$, argument $\arg\lambda(\mathfrak{p}\mathfrak{a}_0)\in[\varphi_1,\varphi_2]$, and $N(\mathfrak{p})\equiv m\pmod M$ is asymptotic to $$A(m,M,C,\mathfrak{a})\frac{(\varphi_2-\varphi_1)y}{2\pi h_{\mathfrak{a}}\varphi(M)\log x}\left(1+O(\tfrac{1}{\log x})\right)$$ under the same ranges as the angular equidistribution theorem of [1]. The constant $A(m,M,C,\mathfrak{a})$ is defined as a sum over characters $\phi$ modulo $M$ and $\psi$ of the ray class group satisfying $\phi(N(\mathfrak{p}))\psi(\mathfrak{p})=1$ for almost all prime ideals; Section 2.6 shows this sum equals $[f_{L_2}:\mathbb{Q}]$ when the Frobenius restrictions of $C$ and $m$ agree in $L_2=K(\mathfrak{a})\cap K(\zeta_M)$, and $0$ otherwise. This gives a fully explicit arithmetic constant in place of the implicit density in the earlier theorem.

Load-bearing premise

The proof's load-bearing step, in Section 2.7, is that after translating the congruence and class conditions into $s$ ideal classes modulo a larger modulus $\mathfrak{b}$, the angular distribution of those classes agrees with the original one up to a fixed rotation and with the same total arc length, so Theorem 1.1 applies with the constant unchanged; if a mismatch occurs, for example because the unit-angle function changes from $\mathfrak{a}$ to $\mathfrak{b}$ so that intervals split or rotate, the computed coefficient $A$ would not match the true count.

Editorial extensions

If this is right

The asymptotic makes angular equidistribution quantitative for arithmetic progressions: for example, the Gaussian primes with norm $p\equiv1\pmod8$ or $p\equiv5\pmod8$ have explicit densities in any angular sector.
Theorem 1.3 gives a limit for the ratio of coordinate power sums over primes $p\equiv m\pmod M$ represented by a primitive positive definite binary quadratic form; Theorem 3.1 extends the limit to bivariate polynomials and homogeneous functions.
The constant $A$ is computable in finite terms: it vanishes exactly when the ray class and the residue class are incompatible, and otherwise equals the degree $[f_{L_2}:\mathbb Q]$ of a subfield of $\mathbb Q(\zeta_M)$.
The numerical data for $x^2+y^2$ and $x^2+xy+y^2$ show the two residue-class ratios oscillating and crossing repeatedly; the paper records as Conjecture 4.1 that the differences change sign infinitely often, and as Conjecture 4.2 that the counting differences $D_1,D_2$ are negatively biased.

Reading between the lines

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

Extension: the same Frobenius-translation technique should apply to other number fields and to several simultaneous congruence conditions, giving angular-congruence asymptotics wherever ray class groups are available; the paper does not state this.
Extension: since $A$ takes only the values $0$ and a field degree, the error term in Theorem 1.2 is plausibly governed by the same Artin $L$-function data as Chebotarev; testing whether the $1/\log x$ error can be sharpened under GRH would connect the result to zero-free regions.
Extension: the symmetric oscillations in $R(N;8,1)$ and $R(N;8,5)$ resemble murmurations; a concrete next test is whether $R(N;M,m)R(N;M,M-m)\to1$ for other moduli $M$, which would indicate a hidden symmetry in the bias.
Extension: Littlewood-type sign changes would follow if the relevant error terms are $O(x^{1/2-\epsilon})$ in an averaged sense; the paper's numerical data do not by themselves prove the conjecture.

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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

My take: Theorem 1.2 genuinely extends Coleman's theorem by adding a norm congruence condition, and it gives an explicit arithmetic coefficient A(m,M,C,a). The coefficient is not a black box; Section 2.6 shows it is either 0 or [f_{L2}:Q] via Chebotarev. That is a real new result, and the application to Elsholtz–Harman with fixed residue classes of representable primes is a sensible payoff.

The proof is mostly sound. The Hecke L-function machinery in Section 2 is standard and executed carefully. Lifting phi and psi to Hecke characters and identifying L(phi,psi,s) up to finitely many factors is correct. The consistency check between the L-function and Chebotarev computations is a nice touch.

The soft spot is exactly where the reader's report points: Section 2.7. The authors say the result is immediate from Theorem 1.1 after translating to classes modulo b. They do not prove that the angular distribution with respect to lambda_a on the original class agrees with the angular distribution with respect to lambda_b on the lifted classes. If g(b) did not divide g(a), intervals in lambda_a would not pull back to intervals in lambda_b of the same total length, and the constant A would change. But here the gap is fillable: a | b follows from K(a) subset K(b), and then the units congruent to 1 mod b form a subgroup of those congruent to 1 mod a, so g(b) | g(a); on each lifted class lambda_a is a fixed rotation composed with an integer power of lambda_b, and the arc length is preserved. The stress-test worry does not land. Still, the paper should show this; as written "the result is immediate" is too quick for the step that fixes the main constant.

The other soft spots are minor. The proof of Theorem 1.3 is a sketch; it relies on the Elsholtz–Harman dissection, and a referee will want the details checked. The conjectures in Section 4 are empirical; they are labeled as conjectures and the numerical evidence is explicit. The Chebyshev-bias part of the title promises a bit more than the paper actually proves, but the authors are upfront about that. The prose has typos but nothing that obscures the math.

Who is this for? Analytic number theorists working on equidistribution of prime ideals, and people who use Coleman-type results for applications to quadratic forms. It deserves a serious referee. With a revision that spells out the Section 2.7 compatibility and expands the application proof, it should be publishable.

Referee Report

2 major / 6 minor

Summary. The paper studies the joint distribution of prime ideals in imaginary quadratic fields with respect to three simultaneous conditions: an ideal class modulo an ideal a, a norm congruence N(p) ≡ m (mod M), and an angular interval for the Coleman-type argument function. The main theorem (Theorem 1.2) asserts that the number of such prime ideals with norm in [x−y, x] satisfies the asymptotic A(m,M,C,a)(φ2−φ1)y/(2π h_a φ(M) log x)(1+O(1/log x)), where A is a character sum that the authors evaluate via class field theory as either 0 or [f_{L2}:Q]. The proof combines Coleman's equidistribution theorem (Theorem 1.1) with Hecke L-function arguments and a Chebotarev-theoretic translation. The paper then applies Theorem 1.2 to limits of power sums of representations by binary quadratic forms (Theorems 1.3 and 3.1) and presents numerical experiments on Chebyshev-bias-like oscillations.

Significance. If Theorem 1.2 is established, it gives a natural and useful refinement of Coleman's theorem: it adds congruence restrictions while preserving the explicit angular density and provides a computable arithmetic coefficient. The Chebotarev-theoretic evaluation of A is elegant and provides an independent check of the character-sum definition. The applications to quadratic forms and the conjectures on sign changes are suggestive. The paper relies on established external results, and the coefficient A is computed from characters rather than fitted, which is a strength. The main potential impact is moderate but genuine in analytic number theory.

major comments (2)

[Section 2.7] The proof of Theorem 1.2 hinges on the assertion, in the last paragraph of Section 2.7, that after translating the conditions p ∈ C, N(p) ≡ m (mod M) to a union of s ideal classes modulo b, 'the result is immediate by Theorem 1.1'. This step fixes the coefficient A, but the required compatibility of the argument functions is not shown. Theorem 1.1 counts primes by the argument of λ_b(p a0'), whereas Theorem 1.2 requires the argument of λ_a(p a0). For the deduction to work, one must prove that on each refined class C_i ⊂ Cl(K,b) there is a constant c_i of modulus 1 and an integer d = g(a)/g(b) such that λ_a(p a0) = c_i λ_b(p a0_i)^d for all p ∈ C_i. This needs (i) g(b) | g(a), which follows from the containment of the unit groups but is not stated, and (ii) a consistent choice of base ideals a0, a0_i so that the ratio a0_i/a0 is principal with a generator satisfying the appropriate congruence. Without such a relation, an interval [φ1, φ2] for λ_a pulls back to a union of intervals for λ_b whose total length need not be φ2 − φ1; if the scaling is not handled, the main term would be off by a factor g(b)/g(a) or by a nonconstant twist. Since the claimed asymptotic depends on the exact value of A computed in Section 2.6, this missing verification is load-bearing and must be supplied in a revision.
[Section 3] The proof of Theorem 1.3 is not actually given: the text states that it is 'a straightforward generalization' of Elsholtz and Harman's method and that 'Now, we have proved Theorem 1.3', but no details of the polar-box dissection are provided. To justify the claimed limit of power sums Σ a_p^k / Σ b_p^k, one must show how Theorem 1.2 is applied to boxes whose angular widths satisfy φ2 − φ1 > x^{−5/24+ε}, sum the resulting asymptotics, and bound the cumulative error. One must also address the possibility that the constant A(m,M,C,a) vanishes for the relevant class, which is not ruled out by the condition that Pm,M ∩ {Q(x,y)} is infinite. Please provide at least an outline of the summation and a statement of the nonvanishing condition.

minor comments (6)

[Section 2.7] The notation 'Denote [L3 : K(b)] by s' appears to be a typo: since the next line uses h_b = [K(b):K] = [K(b):L3][L3:K] = s[L3:K], the quantity s should be [K(b) : L3], not [L3 : K(b)].
[Section 2.6] In the display following the definition of eϕ and ψ, the quantifier '∀σ ∈ Gal(K(a)/K)' should presumably read '∀σ ∈ Gal(L3/K)', since the condition is about the joint action on K(a) and K(ζ_M).
[Section 2.6] The phrase 'the element determined up by' should be 'the element uniquely determined by'.
[General] There are several typos: 'osillations' (Section 1 and Section 4.1) should be 'oscillations'; 'nonnegetive' (Section 3) should be 'nonnegative'; 'principle fractional ideal group' (Section 2.2) should be 'principal fractional ideal group'.
[Section 2.1] In the proof of Corollary 2.2, the condition on ψ in the final sum is written as 'ψ,ψ(p)=1'; this should be stated as ψ(p)=1 for almost all p, which in fact means ψ is the trivial character.
[Section 2.2] The paper uses λ both for the absolute argument function defined in Section 2.2 and for the modulus-dependent argument function in Theorem 1.1. This is a source of confusion; please distinguish the two, for instance by writing λ_f for the ray-class argument function.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 1.2 is derived from external theorems (Coleman, Serre, Hecke L-function theory, class field theory), and the arithmetic coefficient A is computed explicitly, not fitted.

full rationale

The derivation chain is self-contained and non-circular. Theorem 1.2 is obtained in §2.7 by translating the joint conditions (ideal class mod a, norm congruent to m mod M) into a union of ideal classes modulo a larger modulus b via class field theory, then applying Coleman's Theorem 1.1 to each class. The coefficient A(m,M,C,a) is not inferred from the target count; it is defined in Theorem 1.2 and evaluated explicitly in §2.6 by Chebotarev density theory as either 0 or [f_{L2}:Q], depending on compatibility of the Frobenius conditions. Theorems 2.1 and 2.4 are proved from Hecke L-function holomorphy plus Serre's equidistribution criterion (Lemmas 2.7–2.9), with λ lifted to a Hecke character by an explicit local definition; the L-function computations show the relevant sums are o(x/log x) rather than assuming them. There are no fitted parameters, no self-citations, and no appeal to a uniqueness theorem from the present authors. The §2.7 step the reader flags—compatibility of the angular parameters λ_a and λ_b—is a potential gap in justification, but it is not circularity: even if the reduction were incomplete, no step defines a quantity in terms of the target result or predicts an input. The numerical and conjectural sections do not feed back into the proof.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard analytic number theory (Hecke L-functions, Chebotarev) and on Coleman's theorem. No parameters are fitted to data. No new entities are introduced.

assumptions (5)

standard math Serre's equidistribution criterion (Lemma 2.8) and the Weyl criterion (Lemma 2.7) from [9].
Used as the main tool to convert analytic pole data into equidistribution statements in Theorems 2.1 and 2.4.
standard math Hecke L-functions extend meromorphically to the whole plane and have no zeros on Re(s)=1, with poles only for trivial characters (Lemma 2.9 from [5]).
This is the analytic backbone of the proof of Theorem 2.1 and Theorem 2.4.
domain assumption Coleman's Theorem 1.1 on argument equidistribution of prime ideals in imaginary quadratic fields for short intervals [1].
Used as the input in Section 2.7 to get the short-interval version, Theorem 1.2.
standard math Global class field theory, including the Artin map and the existence of a modulus b with L3 ⊂ K(b) (a Kronecker-Weber-type theorem for abelian extensions of number fields).
Invoked in Sections 2.6 and 2.7 to translate congruence and class conditions on prime ideals into Frobenius conditions in an abelian extension.
standard math The global norm map on ideles from K to Q is surjective, so that Dirichlet characters can be inflated to Hecke characters of K via the norm.
Needed in the proof of Theorem 2.1 to lift ϕ to the Hecke character eϕ.

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Pith. "Pith review of The asymptotic estimation of prime ideals in imaginary quadratic fields and Chebyshev's bias." pith.science (2026). https://pith.science/paper/GCM2OCG6

@misc{pith2026241214792,
  author       = {Pith},
  title        = {Pith review of: The asymptotic estimation of prime ideals in imaginary quadratic fields and Chebyshev's bias},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GCM2OCG6}},
  note         = {Machine review of arXiv:2412.14792}
}

read the original abstract

We study the asymptotic estimation of prime ideals that satisfy certain congruence and argument conditions in imaginary quadratic fields. We also discuss the phenomenon of Chebyshev's bias in the distribution of prime ideals among different residue classes.

Figures

Figures reproduced from arXiv: 2412.14792 by the authors.

**Figure 2.** F(N; 12, 1) and F(N; 12, 7) for N ∈ [100, 1 × 106 ]: F(N; 12, 1) is marked blue, F(N; 12, 7) is marked red. We only mark points with 100|N. We are interested in the phenomenon of oscillation and repeated intersections of two function as is shown in the above two figures. Now we define another function, which generates a graph that preserves oscillatory behavior and intersections while exhibiting nice symmetry. This … view at source ↗

**Figure 3.** [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 3.** Figure 3: R(N; 8, 1) and R(N; 8, 5) for N ∈ [100, 5 × 106 ]: R(N; 8, 1) is marked blue, R(N; 8, 5) is marked red. We only mark points with 100|N. This graph show very nice symmetry since F(N; 8, 1), F(N; 8, 5) and F(N) have the same limit as N → ∞. These three figures above all …

**Figure 4.** Figure 4: D1(x) and D2(x) for x ∈ [2, 1.5 × 106 ]: D1(x) is marked blue, D2(x) is marked red. Based on [PITH_FULL_IMAGE:figures/full_fig_p020_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

11 extracted references · 10 canonical work pages

[1]

M. D. Coleman, The distribution of points at which binary quadratic forms are prime, Proceedings of The London Mathematical Society (3) 61 (1990) 433–456

work page 1990
[2]

M. D. Coleman, The distribution of points at which norm-forms are prime, Journal of Number Theory 41 (1992) 359–378

work page 1992
[3]

Devin, Discrepancies in the distribution of gaussian primes, arXiv:2105.02492

L. Devin, Discrepancies in the distribution of gaussian primes, arXiv:2105.02492

work page arXiv
[4]

Elsholtz and G

C. Elsholtz and G. Harman, On conjectures of T. Ordowski and Z. W. Sun con- cerning primes and quadratic forms, Analytic number theory , 65–81. Springer, Cham, 2015

work page 2015
[5]

K. Kato, N. Kurokawa and T. Saito, Number theory. 1, volume 186 of Translations of Mathematical Monographs . American Mathematical Society, Providence, RI,

work page
[6]

Oliver and A

K-H Lee, T. Oliver and A. Pozdnyakov, Murmurations of Dirichlet characters, arXiv:2307.00256

arXiv
[7]

E Littlewood, Sur la distribution des nombres premiers, CR Acad

J. E Littlewood, Sur la distribution des nombres premiers, CR Acad. Sci. Paris 158 (1914) 1869–1872

work page 1914
[8]

Rubinstein and P

M. Rubinstein and P. Sarnak, Chebyshev’s bias, Experimental Mathematics 3(3) (1994) 173–197

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[9]

J. P. Serre, Abelian l-adic representation and elliptic curves, Advanced book clas- sics, 1989

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

M. D. Coleman, The distribution of points at which binary quadratic forms are prime, Proceedings of The London Mathematical Society (3) 61 (1990) 433–456

work page 1990

[2] [2]

M. D. Coleman, The distribution of points at which norm-forms are prime, Journal of Number Theory 41 (1992) 359–378

work page 1992

[3] [3]

Devin, Discrepancies in the distribution of gaussian primes, arXiv:2105.02492

L. Devin, Discrepancies in the distribution of gaussian primes, arXiv:2105.02492

work page arXiv

[4] [4]

Elsholtz and G

C. Elsholtz and G. Harman, On conjectures of T. Ordowski and Z. W. Sun con- cerning primes and quadratic forms, Analytic number theory , 65–81. Springer, Cham, 2015

work page 2015

[5] [5]

K. Kato, N. Kurokawa and T. Saito, Number theory. 1, volume 186 of Translations of Mathematical Monographs . American Mathematical Society, Providence, RI,

work page

[6] [6]

Oliver and A

K-H Lee, T. Oliver and A. Pozdnyakov, Murmurations of Dirichlet characters, arXiv:2307.00256

arXiv

[7] [7]

E Littlewood, Sur la distribution des nombres premiers, CR Acad

J. E Littlewood, Sur la distribution des nombres premiers, CR Acad. Sci. Paris 158 (1914) 1869–1872

work page 1914

[8] [8]

Rubinstein and P

M. Rubinstein and P. Sarnak, Chebyshev’s bias, Experimental Mathematics 3(3) (1994) 173–197

work page 1994