{"id":"37148592-1cc6-4e06-9260-ed451c7efbb7","arxiv_id":"2506.13883","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For Heegner points of different discriminants on the modular surface, the hyperbolic lattice point counting error improves from X^{2/3} to X^{2/3}/(log X)^{1/6}.","lead":"Selberg's 1977 error term in the hyperbolic circle problem is improved logarithmically for the first time, for pairs of Heegner points attached to different imaginary quadratic fields. The reusable ingredient is a new fractional moment estimate for products of two Rankin-Selberg L-functions.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; the cited Selberg orthogonality estimates are worth a direct check.","rationale":"The reader's conditional verdict is reasonable. The single most load-bearing input is indeed the external Selberg orthogonality estimate in Lemma 4.5, which is not proved in the paper; the final exponent depends on it. However, this is a standard Rankin-Selberg consequence for self-contragredient weight-1 theta series, and the cited references are apt. The reader's second stated concern, the missing d == 1 mod 4 hypothesis in Theorem 1.1, is not a genuine overclaim: by the paper's own definition in Section 2.1, a discriminant must be congruent to 0 or 1 mod 4, and a negative squarefree integer cannot be 0 mod 4, so every negative squarefree discriminant is automatically 1 mod 4. The proof of Theorem 1.1 spells this out explicitly. I checked the mollifier argument and the conversion of the fractional moment bound into the spectral sum and then into the circle-problem error term; the chain from Lemma 4.5 to Theorem 1.2, to Proposition 5.1, to Theorem 1.1 is internally consistent, and I found no fitted parameters or circular reasoning. The main reason to preserve the conditional verdict is that the exact applicability of [39,40,41,1] to the mixed cuspidal/Eisenstein cases in Lemma 4.5 should be confirmed before unconditional acceptance; the rest of the proof reads as sound.","tokens_in":35869,"tokens_out":34056,"duration_ms":343086,"concrete_test":"Check the exact hypotheses of [39, Corollary 1.5] and [40, Corollary 1.5] against the class-group theta-series L-functions f_xi (weight 1, level |d|, nebentypus chi_d), including the cases where one factor is a genus character and hence an Eisenstein series; if those references exclude such cases, supply a Rankin-Selberg derivation of Lemma 4.5(1)-(4). Also verify that the Selberg orthogonality constant in the genus-genus case B_xi,xi' is correctly computed when the trivial factorizations are included.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No load-bearing internal flaw found. The logarithmic saving in Theorem 1.2 ultimately rests on Lemma 4.5, which imports Selberg-orthogonality bounds from [39,40,41,1]. Those bounds set the exponent theta >= 1/4 that becomes the 1/6 saving in Theorem 1.1; if the cited results do not apply to the exact class-group theta-series setting, the exponent would fail. The estimates are standard Rankin-Selberg consequences for self-contragredient weight-1 theta series, and Ramanujan is known here, so I do not see a defect. The apparent d == 1 mod 4 restriction is also automatic: under the paper's definition in Section 2.1 a discriminant is b^2 - 4ac with congruence 0 or 1 mod 4, and a negative squarefree discriminant cannot be 0 mod 4, so it must be 1 mod 4. Thus the theorem statement is not an overclaim. The analytic sections (Kuznetsov transform, mollifier iteration, pre-trace formula) appear coherent, and I found no circular dependence on the target theorem.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves that for the full modular group and for Heegner points z_d, z_d' attached to distinct negative squarefree fundamental discriminants d, d', the hyperbolic circle-counting function satisfies N(X;z_d,z_d') = (2π/vol(Γ\\H)) X + O(X^{2/3}/(log X)^{1/6}). This is the first unconditional improvement over Selberg's classical O(X^{2/3}) bound in this setting. The strategy is to combine Waldspurger's formula, which expresses |φ_j(z_d)|^2 through central values of Rankin-Selberg L-functions L(1/2, φ_j × f_ξ), with a new twisted first-moment estimate proved by Kuznetsov's trace formula (Theorem 3.1) and a fractional-moment estimate for products of such central values (Theorem 1.2). A pre-trace formula argument (Proposition 5.1) converts the resulting spectral bound into the counting-function error term. The paper also derives applications to counting binary quadratic forms and a second-moment bound for the error term.","tokens_in":35895,"tokens_out":25624,"duration_ms":252257,"significance":"If the proof is correct, this is a genuine breakthrough: it removes the forty-year-old barrier at O(X^{2/3}) for the hyperbolic circle problem in a natural arithmetic family, and it introduces a fractional-moment technique for Rankin-Selberg L-functions attached to class-group theta series that is likely to have further applications. The proof is unusually detailed: the Kuznetsov transform estimates, the mollifier iteration, and the spectral-to-counting conversion are all written out, and the external inputs are explicitly identified. The main potential risk identified by the stress test is the reliance of Lemma 4.5 on Selberg orthogonality results of Liu-Ye and Avdispahić-Smajlović; on reading the paper, I do not find a defect here, since the needed sums are standard Rankin-Selberg consequences and the citations are to the appropriate general theorems. I also see no circularity: the target error bound is never assumed, and no parameters are fitted to the desired output.","major_comments":[],"minor_comments":[{"comment":"Theorem 1.1 states that d,d' are 'different squarefree discriminants' but does not state the standing assumption d≡1 mod 4 introduced at the start of Section 2.1. For a negative squarefree fundamental discriminant this congruence is automatic, but the theorem should either say 'fundamental discriminants' or explicitly include d,d'≡1 mod 4 so that its hypotheses visibly match those of Theorem 1.2 and Proposition 5.1.","section":"Theorem 1.1 and Section 2.1"},{"comment":"Lemma 4.5 is the load-bearing input that sets the exponent θ≥1/4 in (4.40), and its proof is only a citation to [39,40,41,1]. Because the theta series f_ξ of Section 2.3 are weight-one forms with nebentypus, may be non-self-contragredient, and include genus characters for which the form is Eisenstein rather than cuspidal, the manuscript should state precisely which theorem in the cited papers applies to each of the four assertions, or give a short Rankin-Selberg verification. I believe the standard results do cover the situation, but the current presentation leaves the verification to the reader at the single most delicate point of the argument.","section":"Section 4, Lemma 4.5"},{"comment":"The text defines C1>100 and also says it is 'sufficiently large', but the inequality in (4.41) requires C1 to exceed a much larger absolute constant, depending on the numerical constants in (4.34). Please make the quantifier uniform by stating once that C1 is chosen larger than all absolute constants appearing in the subsequent estimates, so that (4.41) is visibly valid.","section":"Equations (4.9) and (4.41)"},{"comment":"There is a duplicated word in 'To prove Theorem 3.1 we we will use the same-sign Kuznetsov trace formula'; this should be corrected to 'we will use'.","section":"Section 3.2, first line"}],"recommendation":"minor_revision","confidential_remarks":"The main residual risk is Lemma 4.5: if the cited Selberg orthogonality results do not literally cover the weight-one, non-self-contragredient theta-series setting, the logarithmic saving in Theorem 1.1 would collapse. I view this as unlikely, since the relevant Rankin-Selberg L-functions are standard and Ramanujan is known in weight one, but it would be prudent for the handling editor to have one referee confirm the applicability of [39,40,41,1] before publication. All other technical points that I checked appear sound."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the short version: this paper proves the first unconditional improvement of Selberg's O(X^{2/3}) bound in the hyperbolic circle problem, for Heegner points of different discriminants, with a logarithmic saving of (log X)^{-1/6}. The main new ingredient is a fractional moment estimate for products of two Rankin–Selberg L-values at the central point, built on a twisted first moment proved via Kuznetsov. That part is genuinely new and likely useful beyond this problem.\n\nThe paper is well written and the proof is laid out in unusual detail. I found no fitted parameters and no circular dependence on the target theorem. The d ≡ 1 mod 4 restriction that might look like an overclaim is a non-issue: for a squarefree negative discriminant, the congruence d ≡ 0 mod 4 is impossible, so the statement of Theorem 1.1 is fine as written.\n\nThe one thing I would want checked before trusting the exponent is Lemma 4.5, which imports Selberg orthogonality estimates for coefficients of theta series from [39,40,41,1]. Those estimates set the size of the saving (the 1/6 exponent). They are standard in the self-contragredient setting, and Ramanujan is known here, so I do not think there is a real flaw, but it is the main external input and worth a direct look rather than a hand-wave.\n\nThe applications to quadratic form counting (Theorems 1.3 and 1.4) are clean corollaries, and Corollary 1.5 improving the mean-square to (log X)^{3/4} is a nice extra. The self-citations appear as context, not as load-bearing inputs.\n\nBottom line: this deserves a serious referee. It is a strong paper with a clear new result and a largely self-contained proof. If I were the editor, I would send it to a good analytic number theorist, asking them to verify the stationary phase estimates in Section 3 and the cited Selberg orthogonality input.","headline":"First unconditional improvement of Selberg's 40-year-old bound for Heegner points of different discriminants; the proof is detailed, honest, and the apparent congruence issue evaporates on inspection.","tokens_in":36598,"tokens_out":2012,"would_cite":true,"duration_ms":20098,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11P21","11N45","11F67","11E45","11M32"],"pacs":[],"model":"deepseek-v4-flash","headline":"For Heegner points of different discriminants, the hyperbolic circle problem's error term is improved for the first time since 1977.","keywords":["hyperbolic circle problem","Heegner points","modular group","Rankin-Selberg L-functions","fractional moments","toric-period formula","binary quadratic forms","Selberg's bound"],"falsifier":"For a non-genus class group character $\\xi$ of $\\mathbb{Q}(\\sqrt d)$ and any $\\xi'$ of $\\mathbb{Q}(\\sqrt{d'})$, numerically compute $S(x)=\\sum_{p\\le x}\\lambda_\\xi(p)\\lambda_{\\xi'}(p)/p$ for $x$ up to, say, $10^9$; if $|S(x)|$ is not eventually bounded (for instance, if it grows like a positive power of $\\log x$), Lemma 4.5(1) fails and the fractional-moment saving (1.5) is false, so the error term in Theorem 1.1 would not follow from this argument. Conversely, one could test Theorem 1.2 directly by evaluating the $L$-value sum for $T$ around $10^4$ and checking whether its ratio to $T^2/(\\log T)^{1/4}$ stays bounded.","tokens_in":35518,"feed_emoji":"📐","tokens_out":9171,"duration_ms":83107,"temperature":0.7,"pith_summary":"The paper establishes a new error-term estimate for the counting function that records how many images of one Heegner point under the modular group fall within a given hyperbolic distance of another Heegner point, where the two points have different negative squarefree discriminants. It proves that this count equals the expected main term $(2\\pi/\\operatorname{vol}(\\Gamma\\backslash\\mathbb{H}))X$ with error $O(X^{2/3}/(\\log X)^{1/6})$, a logarithmic improvement over the 1977 bound $O(X^{2/3})$ that had never been beaten for any cofinite group or pair of points. The route goes through a new fractional moment estimate for central values of Rankin--Selberg $L$-functions attached to Hecke--Maass forms and $\\theta$ series of imaginary quadratic fields, using a toric-period formula to convert products of eigenfunction values into products of $L$-values. If correct, this is the first unconditional evidence in a concrete arithmetic setting that the conjectured error $O(X^{1/2+\\varepsilon})$ is approachable.","feed_headline":"Heegner point pairs beat Selberg's 1977 lattice-counting bound","feed_subtitle":"First logarithmic gain in the hyperbolic circle problem: error drops by (log X)^{1/6}.","key_machinery":"The engine is the combination of a toric-period formula with fractional moments. The toric-period formula expresses $L(1/2,\\phi_j\\times f_\\xi)/L(1,\\operatorname{sym}^2\\phi_j)$ as a constant multiple of $|\\sum_{a\\in\\mathrm{Cl}_K}\\xi(a)\\phi_j(z_a)|^2$, so pointwise values of Hecke--Maass forms at Heegner points are controlled by central Rankin--Selberg $L$-values twisted by class group characters. The paper then estimates a fractional moment of the product of two such $L$-values using mollifiers $M_r(j,\\xi,\\xi')$ built from short Euler-product factors and the elementary inequality $2\\sqrt{LL'}\\le LM(M')^{-1}+L'M'M^{-1}$. Applying an iterated mollifier decomposition to split the spectral window into sets where the mollifiers expand into short Dirichlet polynomials, together with a new twisted first-moment asymptotic proved through the Kuznetsov formula, yields Theorem 1.2; the cancellation entering through Selberg-type orthogonality estimates fixes the exponent $\\eta=1/4$ that becomes the $1/6$ in the error term via smoothing with a parameter $\\delta=X^{-1/3}(\\log X)^{-2\\eta/3}$.","core_discovery":"On its own terms, the paper's central claim is Theorem 1.1: for $\\Gamma=\\mathrm{SL}_2(\\mathbb{Z})$ and two Heegner points $z_d,z_{d'}$ with squarefree discriminants $d\\neq d'$, $d,d'<0$, the hyperbolic lattice-point count has main term $(2\\pi/\\operatorname{vol}(\\Gamma\\backslash\\mathbb{H}))X$ and error $O(X^{2/3}/(\\log X)^{1/6})$. The logarithmic saving is produced by Theorem 1.2, a fractional moment estimate: over a spectral window $T