--- title: 'Hyperbolic Circle Problem: Lattice-Point Counting' url: https://www.emergentmind.com/topics/hyperbolic-circle-problem type: topic --- # Hyperbolic Circle Problem: Lattice-Point Counting Searching arXiv for the cited hyperbolic circle problem papers to ground the article in recent literature. arXiv search query: hyperbolic circle problem Heegner points Selberg 2506.13883 The hyperbolic circle problem is the lattice-point counting problem for orbits of a Fuchsian group in the hyperbolic plane. For a finite-volume group \(\Gamma \subseteq PSL_2(\mathbb R)\), points \(z,w \in \mathbb H\), and large radius parameter \(R\), one asks for the asymptotic behavior of the number of orbit points \(\gamma z\) lying in the hyperbolic disc of radius \(R\) centered at \(w\). In one standard normalization, \[ N(z,w,X)=\#\{\gamma\in\Gamma:4u(\gamma z,w)+2\le X\}, \qquad u(z,w)=\frac{|z-w|^2}{4\Im(z)\Im(w)}, \] with \(1+2u=\cosh d(z,w)\) and \(X=2\cosh R\). The problem sits at the intersection of spectral theory, automorphic forms, and analytic number theory, and its central difficulty is the size of the error term in the asymptotic formula for \(N(z,w,X)\). The longstanding benchmark is Selberg’s \(O(X^{2/3})\) bound, while the conjectured optimal error is \(O(X^{1/2+\varepsilon})\). Recent work has produced improvements in averaged settings, local \(L^2\) settings, and, for pairs of distinct Heegner points, the first unconditional pointwise improvement over Selberg’s exponent [1709.03282], [2506.13883]. ## 1. Formulation, normalizations, and classical asymptotics For \(\Gamma=\mathrm{PSL}_2(\mathbb Z)\), the hyperbolic circle problem is commonly phrased as estimating \[ N(z,w,X)=\#\{\gamma\in\Gamma:u(\gamma z,w)+2\le X\}, \] or, equivalently, the number of orbit points in a hyperbolic disc of radius \(R\) with \(X=2\cosh R\). For the full modular group in the \(\mathrm{SL}_2(\mathbb Z)\) normalization used in the Heegner-point setting, \[ N(X;z,w)=\#\{\gamma\in\Gamma:4u(z,\gamma w)+2\le X\}. \] The literature quoted here therefore contains both \(\Gamma=\mathrm{PSL}_2(\mathbb Z)\) and \(\Gamma=\mathrm{SL}_2(\mathbb Z)\), and correspondingly writes the main term either as \[ \frac{\pi X}{\mathrm{vol}(\Gamma\backslash\mathbb H)} \quad\text{or}\quad \frac{2\pi}{\mathrm{vol}(\Gamma\backslash\mathbb H)}X. \] Selberg proved that, for fixed \(z,w\), \[ N(X;z,w)=\frac{2\pi}{\mathrm{vol}(\Gamma\backslash\mathbb H)}X+O(X^{2/3}) \] in the \(\mathrm{SL}_2(\mathbb Z)\) normalization, and equivalently \[ N(z,w,X)=\frac{\pi X}{\mathrm{vol}(\Gamma\backslash\mathbb H)}+O(X^{2/3}) \] in the \(\mathrm{PSL}_2(\mathbb Z)\) normalization [2506.13883], [1610.09393]. For a general finite-volume Fuchsian group, one writes more generally \[ N(z,w,X)=\mathcal M(z,w,X)+O_{z,w,\Gamma,\varepsilon}(X^{2/3+\varepsilon}), \] where \(\mathcal M(z,w,X)\) is a spectral main term [1709.03282]. The conjectural best error term is \(O(X^{1/2+\varepsilon})\). That conjecture is supported by mean-value results, but for fixed points it resisted improvement for decades. A central fact emphasized by several recent papers is that gains below the \(2/3\) barrier have so far required either averaging, stronger norms, or additional arithmetic structure [2506.13883], [2403.16113]. ## 2. Spectral formulation and the origin of the \(2/3\) barrier The standard analytic approach is spectral. For an orthonormal basis \(\{\phi_j\}\) of Hecke–Maass cusp forms with spectral parameters \(t_j\), Selberg’s argument reduces the error term to control of spectral sums. In the Heegner-point work, the relevant object is the spectral exponential sum \[ \mathfrak S(T,X):=\sum_{t_j\le T}X^{it_j}\phi_j(z)\overline{\phi_j(w)}. \] The trivial bound \(\mathfrak S(T,X)\ll T^2\) is sufficient for the classical \(O(X^{2/3})\) error term [2506.13883]. For general Fuchsian groups, Biró’s local-average treatment rewrites the counting function as \[ N(z,z,X)=\sum_{\gamma\in\Gamma}k(u(z,\gamma z)), \qquad k(v)=\mathbf 1_{[0,x]}(v), \quad x=\frac{X-2}{4}, \] and decomposes \[ k(v)=k^*(v)+(k(v)-k^*(v)). \] The smoothed part is handled by the Selberg/Harish-Chandra transform and spectral expansion, \[ \sum_{\gamma}k^*(u(z,\gamma z)) = \sum_{j=0}^{\infty}h_{k^*}(t_j)|u_j(z)|^2+\text{(continuous spectrum)}, \] whereas the singular remainder is controlled by a generalized Selberg trace formula that expresses pairings with automorphic eigenfunctions as hyperbolic, elliptic, and parabolic contributions [1709.03282]. This framework clarifies why the classical barrier is robust. Generic control from Weyl law, Cauchy, smoothing, and trace formula technology yields \(2/3\), while sharper exponents demand extra cancellation in spectral or arithmetic data. The recent literature splits accordingly between averaging arguments and arithmetic specializations. ## 3. Averaging, local means, and norm-based improvements A substantial line of progress comes from replacing fixed-point counting by weighted averages. For \(\Gamma=\mathrm{PSL}_2(\mathbb Z)\), averaging over Heegner points of discriminant \(D\) yields \[ \frac{1}{h(D)}\sum_{z\in\mathcal A_D}f(z) \left( N(z,z,X)-\frac{\pi X}{\mathrm{vol}(\Gamma\backslash\mathbb H)} \right) = O\!\left(X^{7/12+\varepsilon}+X^{4/5+\varepsilon}D^{-1/165+\varepsilon}\right), \] for smooth compactly supported non-negative \(f\). If \(D>X^{11/2+\varepsilon}\), this beats Selberg’s exponent, and under the Lindelöf conjecture for twists, the sup-norm conjecture, and bounds on spectral exponential sums one obtains \[ O\!\left(X^{1/2+\varepsilon}+X^{4/5+\varepsilon}D^{-1/10+\varepsilon}\right), \] which is smaller than Selberg’s bound for \(D>X^{4/3+\varepsilon}\) and reaches \(O(X^{1/2+\varepsilon})\) for \(D>X^3\) [1610.09393]. A different averaging result holds for arbitrary finite-volume Fuchsian groups. For a smooth compactly supported weight \(f\) on a fundamental domain \(\mathcal F\), \[ N_f(X):=\int_{\mathcal F}f(z)N(z,z,X)\,d\mu(z) \] satisfies \[ N_f(X) = \int_{\mathcal F} f(z)\sum_{0\le\lambda_j<1} \frac{\Gamma(s_j-\tfrac12)\Gamma(s_j+\tfrac12)}{\Gamma(2s_j)} |u_j(z)|^2X^{s_j}\,d\mu(z) + O_{f,\Gamma,\varepsilon}(X^{5/8+\varepsilon}), \] improving the exponent \(2/3\) to \(5/8\) for any finite-volume Fuchsian group [1709.03282]. Local \(L^2\)-type results form a third regime. For \(\Gamma=PSL_2(\mathbb Z)\), \(F\) the standard fundamental domain, and compact \(S\subset F\), \[ \int_S|N(z,z,X)-3X|^2\,d\mu(z) = O_{S,\varepsilon}(X^{9/7+\varepsilon}), \] which corresponds to a local mean-square error exponent \(e^{(9/14+\varepsilon)R}\) in the radius variable. This is better than the pointwise \(e^{2R/3}\) bound, though weaker than the best local \(L^1\)-average result [2403.16113]. Conditionally on a twisted Linnik-Selberg-type conjecture for sums of Salié sums, this local \(L^2\) exponent can be improved further to some \(c<9/14\) [2604.11205]. These statements concern different norms and averaging procedures, so the exponents are not directly comparable term-by-term. What they do show, collectively, is that averaging is a consistent mechanism for breaking the classical \(2/3\) threshold. | Setting | Quantity | Bound | |---|---|---| | Fixed \(z,w\) | Pointwise error | \(O(X^{2/3})\) | | Heegner average | Averaged error | \(O(X^{7/12+\varepsilon}+X^{4/5+\varepsilon}D^{-1/165+\varepsilon})\) | | Local average, general \(\Gamma\) | Weighted \(L^1\)-average | \(O(X^{5/8+\varepsilon})\) | | Local square mean | \(\int_S |N(z,z,X)-3X|^2\,d\mu(z)\) | \(O(X^{9/7+\varepsilon})\) | | Distinct Heegner points | Pointwise error | \(O(X^{2/3}/(\log X)^{1/6})\) | ## 4. Heegner points and the first unconditional pointwise improvement The most striking recent advance is the pointwise result for pairs of distinct Heegner points \(z_d,z_{d'}\) attached to distinct negative, squarefree discriminants \(d,d'\). For \(\Gamma=\mathrm{SL}_2(\mathbb Z)\), \[ N(X;z_d,z_{d'}) = \frac{2\pi}{\mathrm{vol}(\Gamma\backslash\mathbb H)}X + O\!\left(\frac{X^{2/3}}{(\log X)^{1/6}}\right), \] with the implied constant depending on \(d\) and \(d'\). This is presented as the first unconditional improvement to Selberg’s exponent in any setting, and it depends essentially on the arithmetic structure of Heegner points rather than on generic geometry [2506.13883]. The central spectral improvement is \[ \sum_{t_j\le T}\phi_j(z_d)\overline{\phi_j(z_{d'})} \ll \frac{T^2}{(\log T)^{1/4}}. \] The arithmetic input is Waldspurger’s formula, which relates values of Maass forms at Heegner points to central values of Rankin–Selberg convolutions: \[ \frac{L(\tfrac12,\phi_j\times f_\xi)}{L(1,\operatorname{sym}^2\phi_j)} = c_d^{-1} \left| \sum_{\mathfrak a\in\mathrm{Cl}_K}\xi(\mathfrak a)\phi_j(z_{\mathfrak a}) \right|^2. \] By orthogonality, this gives bounds for \(|\phi_j(z)|\) in terms of square roots of central \(L\)-values [2506.13883]. A principal innovation is a fractional moment estimate for twisted Rankin–Selberg convolutions. For distinct negative, squarefree \(d,d'\equiv1\pmod4\) and class group characters \(\xi,\xi'\), \[ \sum_{T0} \frac{L(\tfrac12,\phi_j\times f_\xi)\lambda_j(\ell)} {L(1,\operatorname{sym}^2\phi_j)}h(t_j), \] uses Dirichlet polynomial mollifiers in the sense pioneered by Radziwiłł and Soundararajan, and exploits the inequality \[ 2\sqrt{LL'} \le LM(M')^{-1}+L'M'M^{-1}. \] The method iteratively extends logarithmic savings from \((\log\log T)^{-1/4}\) to \((\log T)^{-1/4}\) [2506.13883]. The same arithmetic machinery also yields the second-moment estimate \[ \frac1X\int_X^{2X} \left| N(x;z_d,z_{d'}) - \frac{2\pi x}{\mathrm{vol}(\Gamma\backslash\mathbb H)} \right|^2dx \ll X(\log X)^{3/4}, \] improving previous second-moment results, and has applications to counting pairs of quadratic forms of given discriminants and bounded codiscriminant, as well as to mean values of class numbers of such pairs [2506.13883]. ## 5. Weyl sums, central \(L\)-values, and the arithmetic of averaging The Heegner-point averaging method developed earlier already exhibited the same structural bridge between orbit counts and central \(L\)-values. If \(\mathcal A_D\) is the set of Heegner points of discriminant \(D<0\), then Duke’s theorem gives equidistribution as \(|D|\to\infty\), and the class number satisfies \[ |D|^{1/2-\varepsilon}\ll h(D)\ll |D|^{1/2}\log|D| \] for fundamental negative \(D\) [1610.09393]. Spectrally, the averaged discrepancy is governed by terms of the form \[ \sum_j h^+(t_j) \left( \frac1{h(D)}\sum_{z\in\mathcal A_D}f(z)|u_j(z)|^2 - \int_{\Gamma\backslash\mathbb H}f|u_j|^2 \right), \] together with Eisenstein contributions. The corresponding Weyl sums \[ W_c(D,t_j)=\sum_{z\in\mathcal A_D}u_j(z) \] satisfy a Waldspurger–Zhang-type identity \[ |W_c(D,t_j)|^2 = \sqrt{|D|} \frac{L(u_j\otimes\chi_D,1/2)L(u_j,1/2)} {2L(\mathrm{sym}^2u_j,1)}. \] This places the hyperbolic circle problem directly into the analytic theory of twisted automorphic \(L\)-functions [1610.09393]. The analytic input in this regime includes bounds on spectral exponential sums, sup norms, and subconvexity-type estimates. The summary of known ingredients includes the Sarnak–Luo estimate \[ \sum_{|t_j|\le T}X^{it_j}=O(X^\varepsilon(1+T)^{15/4+\varepsilon}), \] the convexity sup-norm bound \(\|u_j\|_\infty=O(|t_j|^{1/2})\), the Iwaniec–Sarnak improvement \[ \|u_j\|_\infty=O(|t_j|^{1/2-1/12+\varepsilon}), \] and the average sup-norm estimate \[ \frac1T\sum_{1\le t_j\le T}\|u_j\|_\infty^2 = O(T^{2(1/2-1/8)+\varepsilon}). \] These bounds explain why arithmetic averaging over Heegner points is effective: it converts eigenfunction values into central \(L\)-values, where deeper cancellation becomes accessible [1610.09393]. A recurrent misconception is that any improvement below \(2/3\) should automatically transfer to fixed generic points. The Heegner-point papers show the opposite: the decisive bridge is Waldspurger’s formula, and that bridge is unavailable for general \(z,w\). This suggests that the present pointwise breakthrough is specifically arithmetic rather than universal. ## 6. Variance, limiting distributions, and dynamical perspectives Another axis of the subject studies fluctuations of the error term. For fixed \(z,w\), one introduces the normalized error \[ e(s,z,w):=\frac{N(s,z,w)-M(s,z,w)}{e^{s/2}}, \] and its fractional integral \[ e_\alpha(s):=I^\alpha e(s) = \frac1{\Gamma(\alpha)}\int_0^s e(u)(s-u)^{\alpha-1}\,du. \] For any \(0<\alpha<1\), the asymptotic variance of \(e_\alpha(s)\) exists and is finite, with explicit formula \[ \lim_{T\to\infty}\frac1T\int_T^{2T}|e_\alpha(s)|^2\,ds = 2\pi\sum_{t_j>0}' \frac{1}{|\Gamma(\tfrac32+it_j)|^2} |\varphi_j(z)\varphi_j(w)|^2, \] up to the additional continuous-spectrum term in the noncocompact case. Moreover, \(e_\alpha(s)\) admits a limiting distribution for every \(0<\alpha<1\), and for \(\alpha>1/2\) that limiting measure is compactly supported [1512.04779]. This probabilistic viewpoint complements, rather than replaces, pointwise estimates. Fractional integration regularizes spectral coefficients that are too slowly decaying in the unsmoothed error term, and thereby makes variance calculations accessible. The original variance problem for \(e(s)\) itself remains open [1512.04779]. A related dynamical perspective comes from effective equidistribution of large circles and circle arcs on compact hyperbolic surfaces. For translates of circle arcs by arbitrary elements of \(\mathrm{SL}_2(\mathbb R)\), spectral methods of Ratner and Burger yield precise asymptotics for circle averages, exponential rates of equidistribution governed by the spectral gap, and applications to lattice counting in hyperbolic balls following Duke–Rudnick–Sarnak and Eskin–McMullen [2208.07771]. This does not replace the automorphic-spectral formulation of the hyperbolic circle problem, but it situates it within a broader program linking counting, mixing, and homogeneous dynamics. The present frontier is therefore sharply defined. For general finite-volume Fuchsian groups, the fixed-point error term remains \(O(X^{2/3})\). Unconditional improvements are known for local averages, local square means, and Heegner specializations; conditional gains below the current \(9/14\) local \(L^2\) threshold depend on conjectural cancellation in sums of Salié sums; and the first unconditional pointwise gain below \(2/3\) relies on the arithmetic of distinct Heegner points [1709.03282], [2604.11205], [2506.13883]. The conjectured \(O(X^{1/2+\varepsilon})\) bound thus remains open in its classical pointwise form, but the modern literature has substantially clarified which spectral, arithmetic, and dynamical mechanisms can move the problem beyond Selberg’s barrier. Source: https://www.emergentmind.com/topics/hyperbolic-circle-problem