scholarly journals Distribution of the error term for the number of lattice points inside a shifted circle

1993 ◽  
Vol 154 (3) ◽  
pp. 433-469 ◽  
Author(s):  
Pavel M. Bleher ◽  
Zheming Cheng ◽  
Freeman J. Dyson ◽  
Joel L. Lebowitz
Keyword(s):  
Error Term ◽  
Lattice Points

Two classical lattice point problems

1961 ◽  
Vol 57 (4) ◽  
pp. 699-721 ◽  
Author(s):  
K. S. Gangadharan ◽  
A. E. Ingham
Keyword(s):  
Lattice Point ◽  
Error Term ◽  
Divisor Problem ◽  
Lattice Points ◽  
Classical Lattice ◽  
Rectangular Hyperbola ◽  
Number Of Divisors ◽  
Image Position ◽  
Euler's Constant ◽  
Lattice Point Problems

Let r(n) be the number of representations of n as a sum of two squares, d(n) the number of divisors of n, andwhere γ is Euler's constant. Then P(x) is the error term in the problem of the lattice points of the circle, and Δ(x) the error term in Dirichlet's divisor problem, or the problem of the lattice points of the rectangular hyperbola.

scholarly journals On the Gaussian Limiting Distribution of Lattice Points in a Parallelepiped

2016 ◽  
Vol 11 (2) ◽  
pp. 45-89
Author(s):  
Mordechay B. Levin
Keyword(s):  
Algebraic Number ◽  
Error Term ◽  
Main Tool ◽  
Random Variable ◽  
Algebraic Number Field ◽  
Limiting Distribution ◽  
Lattice Points ◽  
Point Problem ◽  
Lattice Point Problem ◽  
Totally Real

AbstractLet Γ ⊂ ℝs be a lattice obtained from a module in a totally real algebraic number field. Let ℛ(θ, N) be the error term in the lattice point problem for the parallelepiped [−θ1N1, θ1N1] × . . . × [−θs Ns, θs Ns]. In this paper, we prove that ℛ(θ, N)/σ(ℛ, N) has a Gaussian limiting distribution as N→∞, where θ = (θ1, . . . , θs) is a uniformly distributed random variable in [0, 1]s, N = N1 . . . . Ns and σ(ℛ, N) ≍ (log N)(s−1)/2. We obtain also a similar result for the low discrepancy sequence corresponding to Γ. The main tool is the S-unit theorem.

scholarly journals SECOND MOMENTS IN THE GENERALIZED GAUSS CIRCLE PROBLEM

2018 ◽  
Vol 6 ◽  
Author(s):  
THOMAS A. HULSE ◽  
CHAN IEONG KUAN ◽  
DAVID LOWRY-DUDA ◽  
ALEXANDER WALKER
Keyword(s):  
Error Term ◽  
Power Saving ◽  
Lattice Points ◽  
Dimensional Sphere ◽  
Mean Square ◽  
Circle Problem ◽  
Second Moments ◽  
The Laplace Transform ◽  
Error Terms ◽  
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.

On two classical lattice point problems

1940 ◽  
Vol 36 (2) ◽  
pp. 131-138 ◽  
Author(s):  
A. E. Ingham
Keyword(s):  
Lattice Point ◽  
Error Term ◽  
Divisor Problem ◽  
Lattice Points ◽  
Classical Lattice ◽  
Rectangular Hyperbola ◽  
Number Of Divisors ◽  
Image Position ◽  
Euler's Constant ◽  
Lattice Point Problems

1. Let r(n) be the number of representations of n as a sum of two squares, d(n) the number of divisors of n, andwhere γ is Euler's constant. Thus P(x) is the error term in the problem of the lattice points of a circle, and Δ(x) the error term in Dirichlet's divisor problem, or the problem of the lattice points of a rectangular hyperbola.

scholarly journals Primitive Points in Rational Polygons

2020 ◽  
Vol 63 (4) ◽  
pp. 850-870
Author(s):  
Imre Bárány ◽  
Greg Martin ◽  
Eric Naslund ◽  
Sinai Robins
Keyword(s):  
Error Term ◽  
Lattice Points ◽  
Summatory Function ◽  
Primitive Lattice Points

AbstractLet ${\mathcal{A}}$ be a star-shaped polygon in the plane, with rational vertices, containing the origin. The number of primitive lattice points in the dilate $t{\mathcal{A}}$ is asymptotically $\frac{6}{\unicode[STIX]{x1D70B}^{2}}\text{Area}(t{\mathcal{A}})$ as $t\rightarrow \infty$. We show that the error term is both $\unicode[STIX]{x1D6FA}_{\pm }(t\sqrt{\log \log t})$ and $O(t(\log t)^{2/3}(\log \log t)^{4/3})$. Both bounds extend (to the above class of polygons) known results for the isosceles right triangle, which appear in the literature as bounds for the error term in the summatory function for Euler’s $\unicode[STIX]{x1D719}(n)$.

scholarly journals Optimal spectral rectangles and lattice ellipses

2013 ◽  
Vol 469 (2150) ◽  
pp. 20120492 ◽  
Author(s):  
Pedro R. S. Antunes ◽  
Pedro Freitas
Keyword(s):  
Boundary Conditions ◽  
Rate Of Convergence ◽  
Error Term ◽  
Dirichlet Boundary ◽  
Unit Area ◽  
Integer Lattice ◽  
Lattice Points ◽  
Computational Method ◽  
Dirichlet Boundary Conditions ◽  
Circle Problem

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.

Circles on lattices and Hadamard matrices

2019 ◽  
pp. 2-9 ◽  
Author(s):  
N. A. Balonin ◽  
M. B. Sergeev ◽  
J. Seberry ◽  
O. I. Sinitsyna
Keyword(s):  
Hadamard Matrices ◽  
Lattice Points ◽  
Practical Significance ◽  
Upper And Lower Bounds ◽  
Problem Size ◽  
Orthogonal Matrices ◽  
Video Information ◽  
Scientific Schools ◽  
Gauss Theorem ◽  
Triangular Numbers

Introduction: The Hadamard conjecture about the existence of Hadamard matrices in all orders multiple of 4, and the Gauss problem about the number of points in a circle are among the most important turning points in the development of mathematics. They both stimulated the development of scientific schools around the world with an immense amount of works. There are substantiations that these scientific problems are deeply connected. The number of Gaussian points (Z3 lattice points) on a spheroid, cone, paraboloid or parabola, along with their location, determines the number and types of Hadamard matrices.Purpose: Specification of the upper and lower bounds for the number of Gaussian points (with odd coordinates) on a spheroid depending on the problem size, in order to specify the Gauss theorem (about the solvability of quadratic problems in triangular numbers by projections onto the Liouville plane) with estimates for the case of Hadamard matrices. Methods: The authors, in addition to their previous ideas about proving the Hadamard conjecture on the base of a one-to-one correspondence between orthogonal matrices and Gaussian points, propose one more way, using the properties of generalized circles on Z3 .Results: It is proved that for a spheroid, the lower bound of all Gaussian points with odd coordinates is equal to the equator radius R, the upper limit of the points located above the equator is equal to the length of this equator L=2πR, and the total number of points is limited to 2L. Due to the spheroid symmetry in the sector with positive coordinates (octant), this gives the values of R/8 and L/4. Thus, the number of Gaussian points with odd coordinates does not exceed the border perimeter and is no less than the relative share of the sector in the total volume of the figure.Practical significance: Hadamard matrices associated with lattice points have a direct practical significance for noise-resistant coding, compression and masking of video information.

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