Gauss Circle Problem Research Articles

Around the Gauss circle problem: Hardy's conjecture and the distribution of lattice points near circles

AbstractHardy conjectured that the error term arising from approximating the number of lattice points lying in a radius‐ disc by its area is . One source of support for this conjecture is a folklore heuristic that uses i.i.d. random variables to model the lattice points lying near the boundary and square root cancellation of sums of these random variables. We examine this heuristic by studying how these lattice points interact with one another and prove that their autocorrelation is determined in terms of a random model. Additionally, it is shown that lattice points near the boundary which are “well separated” behave independently. We also formulate a conjecture concerning the distribution of pairs of these lattice points.

픽의 정리를 활용한 Gauss Circle Problem 해결 및 3차원 도형에서의 확장

In this study, a new solution to the Gauss Circle Problem was presented using Pick’s Theorem. The simplest polygon containing both grid points inside the circle and on the circumference was drawn, and the result was derived by applying Pick’s Theorem to the polygon instead of the circle. He analyzed the existing solution method and inductively proved using Python that the conclusion obtained through the new solution method was within the error range known through the existing solution method, revealing that the new solution method was valid. A two-dimensional circle was expanded into a three-dimensional sphere, and a new solution method was applied to three-dimensional space using Reeve’s Theorem, which extends Pick’s Theorem to three dimensions.

On an analogue of the Gauss circle problem for the Heisenberg group

On an analogue of the Gauss circle problem for the Heisenberg group

Rogers' mean value theorem for S-arithmetic Siegel transforms and applications to the geometry of numbers

We prove higher moment formulas for Siegel transforms defined over the space of unimodular S-lattices in QSd, d≥3, where in the real case, the formulas are introduced by Rogers [20]. As applications, we obtain the random statements of Gauss circle problem for any star-shaped sets in QSd centered at the origin and of the effective Oppenheim conjecture for S-arithmetic quadratic forms.

A twist of the Gauss circle problem by holomorphic cusp forms

A twist of the Gauss circle problem by holomorphic cusp forms

Balanced derivatives, identities, and bounds for trigonometric and Bessel series

Motivated by two identities published with Ramanujan's lost notebook and connected, respectively, with the Gauss circle problem and the Dirichlet divisor problem, in an earlier paper, three of the present authors derived representations for certain sums of products of trigonometric functions as double series of Bessel functions [8]. These series are generalized in the present paper by introducing the novel notion of balanced derivatives, leading to further theorems. As we will see below, the regions of convergence in the unbalanced case are entirely different than those in the balanced case. From this viewpoint, it is remarkable that Ramanujan had the intuition to formulate entries that are, in our new terminology, “balanced”. If x denotes the number of products of the trigonometric functions appearing in our sums, in addition to proving the identities mentioned above, theorems and conjectures for upper and lower bounds for the sums as x→∞ are established.

Multiple Fourier series and lattice point problems

For the multiple Fourier series of the periodization of some radial functions on Rd, we investigate the behavior of the spherical partial sum. We show the Gibbs-Wilbraham phenomenon, the Pinsky phenomenon and the third phenomenon for the multiple Fourier series, involving the convergence properties of them. The third phenomenon is closely related to the lattice point problems, which is a classical theme of the analytic number theory. We also prove that, for the case of two or three dimensions, the convergence problem on the Fourier series is equivalent to the lattice point problems in a sense. In particular, the convergence problem at the origin in two dimensions is equivalent to Hardy's conjecture on Gauss's circle problem.

The Weyl formula for planar annuli

We study the zeros of cross-product of Bessel functions and obtain their approximations, based on which we reduce the eigenvalue counting problem for the Dirichlet Laplacian associated with a planar annulus to a lattice point counting problem associated with a special domain in R2. Unlike other lattice point problems, the one arisen naturally here has interesting features that lattice points under consideration are translated by various amounts and the curvature of the boundary is unbounded. By transforming this problem into a relatively standard form and using classical van der Corput's bounds, we obtain a two-term Weyl formula for the eigenvalue counting function for the planar annulus with a remainder of size O(μ2/3). If we additionally assume that certain tangent has rational slope, we obtain an improved remainder estimate of the same strength as Huxley's bound in the Gauss circle problem, namely O(μ131/208(log⁡μ)18627/8320). As a by-product of our lattice point counting results, we readily obtain this Huxley-type remainder estimate in the two-term Weyl formula for planar disks.

The Laplace transform of the second moment in the Gauss circle problem

The Gauss circle problem concerns the difference $P_2(n)$ between the area of a circle of radius $\sqrt{n}$ and the number of lattice points it contains. In this paper, we study the Dirichlet series with coefficients $P_2(n)^2$, and prove that this series has meromorphic continuation to $\mathbb{C}$. Using this series, we prove that the Laplace transform of $P_2(n)^2$ satisfies $\int_0^\infty P_2(t)^2 e^{-t/X} \, dt = C X^{3/2} -X + O(X^{1/2+\epsilon})$, which gives a power-savings improvement to a previous result of Ivic [Ivic1996]. Similarly, we study the meromorphic continuation of the Dirichlet series associated to the correlations $r_2(n+h)r_2(n)$, where $h$ is fixed and $r_2(n)$ denotes the number of representations of $n$ as a sum of two squares. We use this Dirichlet series to prove asymptotics for $\sum_{n \geq 1} r_2(n+h)r_2(n) e^{-n/X}$, and to provide an additional evaluation of the leading coefficient in the asymptotic for $\sum_{n \leq X} r_2(n+h)r_2(n)$.

Bykovskii-Type Theorem for the Picard Manifold

Abstract We generalise a result of Bykovskii to the Gaussian integers and prove an asymptotic formula for the prime geodesic theorem in short intervals on the Picard manifold. Previous works show that individually the remainder is bounded by $O(X^{13/8+\epsilon })$ and $O(X^{3/2+\theta +\epsilon })$, where $\theta$ is the subconvexity exponent for quadratic Dirichlet $L$-functions over $\mathbb{Q}(i)$. By combining arithmetic methods with estimates for a spectral exponential sum and a smooth explicit formula, we obtain an improvement for both of these exponents. Moreover, by assuming two standard conjectures on $L$-functions, we show that it is possible to reduce the exponent below the barrier $3/2$ and get $O(X^{34/23+\epsilon })$ conditionally. We also demonstrate a dependence of the remainder in the short interval estimate on the classical Gauss circle problem for shifted centres.

Higher moments of distances between consecutive Ford spheres

In previous work the second author derived an asymptotic formula for the sum of the distances between centers of consecutive Ford spheres. In this paper we extend these results by proving asymptotic formulas for higher moments of the distances. Our proofs rely on lattice point counting estimates with error terms coming from the Gauss circle problem.

Dimensional lower bounds for Falconer type incidence theorems

Let $1 \leq k \leq d$ and consider a subset $E\subset \mathbb{R}^d$. In this paper, we study the problem of how large the Hausdorff dimension of $E$ must be in order for the set of distinct noncongruent $k$-simplices in $E$ (that is, noncongruent point configurations of $k+1$ points from $E$) to have positive Lebesgue measure. This generalizes the $k=1$ case, the well-known Falconer distance problem and a major open problem in geometric measure theory. We establish a dimensional lower threshold of $\frac{d(k+1)}{d+2}$ for Falconer type theorems for $k$-simplices. This threshold is nontrivial in the range $d/2 \leq k \leq d$ and is obtained through counting simplices in a standard lattice using results of the Gauss circle problem. Many results on Falconer type theorems have been established through incidence theorems, which generally establish sufficient but not necessary conditions for the point configuration theorems. We also establish a dimensional lower threshold of $\frac{d+1}{2}$ on incidence theorems for $k$-simplices where $k\leq d \leq 2k+1$ by generalizing an example of Mattila. Finally, we prove a dimensional lower threshold of $\frac{d+1}{2}$ on incidence theorems for triangles in a convex setting in every dimension greater than $3$. This last result generalizes work by Iosevich and Senger on distances that was built on a construction by Valtr. The final result utilizes number-theoretic machinery to estimate the number of solutions to a Diophantine equation.

The mean square discrepancy in the circle problem

We study the mean square of the error term in the Gauss circle problem. A heuristic argument based on the consideration of off-diagonal terms in the mean square of the relevant Voronoi-type summation formula leads to a precise conjecture for the mean square of this discrepancy.

Stretching Convex Domains to Capture Many Lattice Points

Abstract We consider an optimal stretching problem for strictly convex domains in $\mathbb{R}^d$ that are symmetric with respect to each coordinate hyperplane, where stretching refers to transformation by a diagonal matrix of determinant 1. Specifically, we prove that the stretched convex domain which captures the most positive lattice points in the large volume limit is balanced: the (d − 1)-dimensional measures of the intersections of the domain with each coordinate hyperplane are equal. Our results extend those of Antunes and Freitas, van den Berg, Bucur and Gittins, Ariturk and Laugesen, van den Berg and Gittins, and Gittins and Larson. The approach is motivated by the Fourier analysis techniques used to prove the classical $\#\{(i,j) \in \mathbb{Z}^2 : i^2 +j^2 \le r^2 \} =\pi r^2 + \mathcal{O}(r^{2/3})$ result for the Gauss circle problem.

SECOND MOMENTS IN THE GENERALIZED GAUSS CIRCLE PROBLEM

The generalized Gauss circle problem concerns the lattice point discrepancy of large spheres. We study the Dirichlet series associated to$P_{k}(n)^{2}$, where$P_{k}(n)$is the discrepancy between the volume of the$k$-dimensional sphere of radius$\sqrt{n}$and the number of integer lattice points contained in that sphere. We prove asymptotics with improved power-saving error terms for smoothed sums, including$\sum P_{k}(n)^{2}e^{-n/X}$and the Laplace transform$\int _{0}^{\infty }P_{k}(t)^{2}e^{-t/X}\,dt$, in dimensions$k\geqslant 3$. We also obtain main terms and power-saving error terms for the sharp sums$\sum _{n\leqslant X}P_{k}(n)^{2}$, along with similar results for the sharp integral$\int _{0}^{X}P_{3}(t)^{2}\,dt$. This includes producing the first power-saving error term in mean square for the dimension-3 Gauss circle problem.

Visible lattice points and the Extended Lindelöf Hypothesis

We consider the number of visible lattice points under the assumption of the Extended Lindelöf Hypothesis. We get a relation between visible lattice points and the Extended Lindelöf Hypothesis. And we also get a relation between visible lattice points over Q(−1) and the Gauss Circle Problem.

On representations of error terms related to the derivatives for some Dirichlet series

In previous papers, we examined several properties of an error term in a certain divisor problem related to the derivatives of the Riemann zeta-function. In this paper, we obtain representations of error terms related to the derivatives of some Dirichlet series, which can be regarded as generalized versions of a Dirichlet divisor problem and a Gauss circle problem. We also give the upper bounds of the error terms in terms of exponent pairs.

A generalised Gauss circle problem and integrated density of states

Counting lattice points inside a ball of large radius in Euclidean space is a classical problem in analytic number theory, dating back to Gauss. We propose a variation on this problem: studying the asymptotics of the measure of an integer lattice of affine planes inside a ball. The first term is the volume of the ball; we study the size of the remainder term. While the classical problem is equivalent to counting eigenvalues of the Laplace operator on the torus, our variation corresponds to the integrated density of states of the Laplace operator on the product of a torus with Euclidean space. The asymptotics we obtain are then used to compute the density of states of the magnetic Schrödinger operator.

On lattice sums and Wigner limits

Wigner limits are given formally as the difference between a lattice sum, associated to a positive definite quadratic form, and a corresponding multiple integral. To define these limits, which arose in work of Wigner on the energy of static electron lattices, in a mathematically rigorous way one commonly truncates the lattice sum and the corresponding integral and takes the limit along expanding hypercubes or other regular geometric shapes. We generalize the known mathematically rigorous two- and three-dimensional results regarding Wigner limits, as laid down in [3], to integer lattices of arbitrary dimension. In doing so, we also resolve a problem posed in [6, Chapter 7]. For the sake of clarity, we begin by considering the simpler case of cubic lattice sums first, before treating the case of arbitrary quadratic forms. We also consider limits taken along expanding hyperballs with respect to general norms, and connect with classical topics such as Gauss's circle problem. Appendix A is included to recall certain properties of Epstein zeta functions that are either used in the paper or serve to provide perspective.

Optimal spectral rectangles and lattice ellipses

We consider the problem of minimizing the k th eigenvalue of rectangles with unit area and Dirichlet boundary conditions. This problem corresponds to finding the ellipse centred at the origin with axes on the horizontal and vertical axes with the smallest area containing k integer lattice points in the first quadrant. We show that, as k goes to infinity, the optimal rectangle approaches the square and, correspondingly, the optimal ellipse approaches the circle. We also provide a computational method for determining optimal rectangles for any k and relate the rate of convergence to the square with the conjectured error term for Gauss's circle problem.