{"id":"0d4f01b7-7b7c-4e14-9140-d9cfd43bf111","arxiv_id":"2412.14792","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Prime ideals in imaginary quadratic fields with prescribed norm residue and argument satisfy an explicit asymptotic formula with a character-theoretic coefficient, generalizing Coleman's equidistribution theorem.","lead":"This paper proves an asymptotic formula for the number of prime ideals in imaginary quadratic fields whose norm lies in a given residue class and whose argument lies in a given interval, generalizing a classical theorem of Coleman. It uses Hecke L-functions and class field theory, and applies the result to study Chebyshev's bias for primes represented by quadratic forms.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The load-bearing step is §2.7's unproved compatibility of λ_a and λ_b: translating the congruence to classes modulo b and applying Theorem 1.1 'immediately' requires that on each lifted class λ_a is a fixed rotation composed with an integer power of λ_b, which the paper never proves.","rationale":"The paper's main theorem rests on a Hecke L-function and Chebotarev argument. Sections 2.1–2.5 are standard and sound: the L-functions are correctly identified as Hecke L-functions, and the pole analysis at s = 1 is correct. Section 2.6 correctly computes the character sum A as 0 or [fL2:Q] via the fiber product of Galois groups. The fragile point is Section 2.7, where Coleman's angular equidistribution is imported after changing the modulus from a to b. The reader flagged this as the weakest assumption, and I agree it is the load-bearing step. However, a close analysis shows the compatibility is plausibly true: λ_a and λ_b are Hecke characters whose infinite components differ by an integer power, so their quotient is finite order and factors through a ray class group; hence it is constant on each lifted class, and the arc length is preserved when summing over the r preimage intervals. Thus the concern is a genuine proof gap rather than a demonstrated counterexample. The verdict CONDITIONAL is appropriate: the paper should expand Section 2.7 to prove the λ-compatibility and the arc-length preservation before unconditional acceptance. I see no basis to change the reader's assessment.","tokens_in":14440,"tokens_out":44060,"duration_ms":345964,"concrete_test":"Prove that θ(p) = λ_a(p a0) · λ_b(p a0')^{-g(a)/g(b)} is a finite-order Hecke character of K, hence constant on each ray class modulo b; also verify g(b) | g(a). Then check that the preimage of any interval [φ1, φ2] under λ_a consists of g(a)/g(b) intervals whose total length is exactly φ2 − φ1. As a numerical cross-check, take K = Q(i), a = (1), M = 8, m = 1, compute primes up to 10^6, and confirm that the count with N(p) ≡ 1 (mod 8) equals (φ2 − φ1)y/(4π log x) for several intervals, matching A = 2.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 2.7 reduces the joint conditions p ∈ C (mod a), N(p) ≡ m (mod M) to a union of s ideal classes modulo a larger modulus b, then claims the count 'is immediate by Theorem 1.1'. This is the step that fixes the constant A(m,M,C,a). Theorem 1.1 counts primes by arg λ_b(p a0'), while Theorem 1.2 asks for arg λ_a(p a0). The proof never shows that, on each lifted class, λ_a(p a0) = χ_i(p) · λ_b(p a0')^{g(a)/g(b)} with χ_i a fixed finite-order character constant on that class. If this holds, an interval of length L in λ_a pulls back to r = g(a)/g(b) intervals each of length L/r, so total arc length is preserved and the coefficient [fL2:Q] follows. If χ_i were not constant on the class, or if g(b) did not divide g(a), the angular measure would be twisted and the constant would change. The words 'the result is immediate' sit exactly at the point where the central theorem's coefficient could be wrong; the required verification is omitted.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":14710,"tokens_out":23606,"duration_ms":171826,"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":[{"comment":"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":"Section 2.7"},{"comment":"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.","section":"Section 3"}],"minor_comments":[{"comment":"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":"Section 2.7"},{"comment":"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":"Section 2.6"},{"comment":"The phrase 'the element determined up by' should be 'the element uniquely determined by'.","section":"Section 2.6"},{"comment":"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":"General"},{"comment":"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":"Section 2.1"},{"comment":"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.","section":"Section 2.2"}],"recommendation":"major_revision","confidential_remarks":"The paper is likely correct in substance, and the main gap in Section 2.7 appears fixable: one needs a lemma establishing the relation λ_a(p a0) = c_i λ_b(p a0_i)^{g(a)/g(b)} on each refined class, together with the observation g(b) | g(a). The Chebotarev computation of A is a strong part of the paper. The application section (Theorem 1.3) is currently too sketchy for a journal publication. I do not see grounds for rejection; the issues are within the scope of a revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nThe 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.\n\nThe 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.\n\nWho 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.","headline":"A real extension of Coleman's theorem with a correct but under-verified key step in Section 2.7; worth serious refereeing.","tokens_in":15183,"tokens_out":15047,"would_cite":true,"duration_ms":118883,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11N05","11K70"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["asymptotic estimation","quadratic forms","Chebyshev's bias","imaginary quadratic fields","prime ideals","argument equidistribution","Hecke L-functions","Chebotarev density"],"falsifier":"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.","tokens_in":14271,"feed_emoji":"🔢","tokens_out":14428,"duration_ms":101956,"temperature":0.7,"pith_summary":"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.","feed_headline":"Prime ideals with fixed residue and angle now have explicit counts","feed_subtitle":"Arithmetic coefficient is zero or a field degree; data show Chebyshev-type bias between residue classes.","key_machinery":"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 $\\lambda(\\mathfrak{a})=(\\xi_{\\mathfrak{a}}/|\\xi_{\\mathfrak{a}}|)^g$.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies Theorem 1.1, the angular equidistribution asymptotic for prime ideals in imaginary quadratic fields that Theorem 1.2 refines by adding a norm congruence condition.","marker":"[1]"},{"why":"Supplies the companion norm-form equidistribution result whose method the paper extends.","marker":"[2]"},{"why":"Supplies the quadratic-form ratio theorem for all representable primes that Theorem 1.3 generalizes to primes in a fixed residue class.","marker":"[4]"},{"why":"Provides the Hecke L-function continuation and pole criterion used to evaluate the character sum A.","marker":"[5]"},{"why":"Provides the character-sum equidistribution lemma applied to the compact group of residue classes times the unit circle.","marker":"[9]"},{"why":"Supplies the sums-of-two-squares discrepancy problem that motivates the separated counting functions D1 and D2.","marker":"[3]"},{"why":"Supplies the classical sign-change theorem for primes 1 and 3 modulo 4 that underlies Conjecture 4.1.","marker":"[7]"},{"why":"Defines the logarithmic-density treatment of Chebyshev's bias used to interpret the numerical data.","marker":"[8]"},{"why":"Supplies the conjectures on representations of primes by quadratic forms that the application section extends.","marker":"[10]"}],"fun_headline_variants":["Prime ideals with fixed residue and angle: explicit constants","Chebyshev bias in imaginary quadratic fields now has exact densities","Explicit arithmetic constant for prime ideals with prescribed angle and mod","Prime ideals obey Chebyshev bias: explicit zero-or-degree constants"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Prime ideals with fixed residue and angle: explicit constants","Chebyshev bias in imaginary quadratic fields now has exact densities","Explicit arithmetic constant for prime ideals with prescribed angle and mod","Prime ideals obey Chebyshev bias: explicit zero-or-degree constants"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000226,"raw_usage":{"total_tokens":1427,"prompt_tokens":862,"completion_tokens":565,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":478,"completion_tokens_details":{"reasoning_tokens":495}},"tokens_in":478,"tokens_out":565,"duration_ms":16619,"temperature":1.0,"reasoning_tokens":495,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T11:55:13.825107+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies Theorem 1.1, the angular equidistribution asymptotic for prime ideals in imaginary quadratic fields that Theorem 1.2 refines by adding a norm congruence condition."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the companion norm-form equidistribution result whose method the paper extends."},{"cited_title":"Elsholtz and G","cited_arxiv_id":null,"evidence_quote":"Supplies the quadratic-form ratio theorem for all representable primes that Theorem 1.3 generalizes to primes in a fixed residue class."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Hecke L-function continuation and pole criterion used to evaluate the character sum A."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the character-sum equidistribution lemma applied to the compact group of residue classes times the unit circle."},{"cited_title":"Devin, Discrepancies in the distribution of gaussian primes, arXiv:2105.02492","cited_arxiv_id":null,"evidence_quote":"Supplies the sums-of-two-squares discrepancy problem that motivates the separated counting functions D1 and D2."},{"cited_title":"E Littlewood, Sur la distribution des nombres premiers, CR Acad","cited_arxiv_id":null,"evidence_quote":"Supplies the classical sign-change theorem for primes 1 and 3 modulo 4 that underlies Conjecture 4.1."},{"cited_title":"Rubinstein and P","cited_arxiv_id":null,"evidence_quote":"Defines the logarithmic-density treatment of Chebyshev's bias used to interpret the numerical data."},{"cited_title":"Conjectures on representations involving primes","cited_arxiv_id":"1211.1588","evidence_quote":"Supplies the conjectures on representations of primes by quadratic forms that the application section extends."}],"review_version":1}