Visible lattice points in the sphere

Abstract

The number of visible (primitive) lattice points in the sphere of radius R is well approximated by 4π 3ζ(3) R 3 . We consider an integral expression involving the error term E * (R), which leads to E * (R) = Ω(R(log R) 1/2 ). This is comparable to what is known in the sphere problem. We can avoid the use of the second power moment (which is in this case unknown) by employing an auxiliary trigonometric series correlated to E * (R). This approach to prove Ω-results seems to be new and could be useful in other problems.

The linking number in systems with Periodic Boundary Conditions

“…For d = 3, the ratio between the number of points and the naive estimate above is bounded, up to constants only depending on , from above by n and from below by n − , for all > 0. More precisely, in [69], we see that…”

The sphere problem and the L-functions

“…It is not difficult to prove that θ 3 1, in fact it is known [24] that S(R) − 4π 3 R 3 is not o R(log R) 1/2 (the same holds for visible points [2], and for general convex bodies with a smaller logarithmic power [19]). In general, upper bounds are conjecturally less precise.…”

On the Characterization of Absentee-Voxels in a Spherical Surface and Volume of Revolution in $${\mathbb Z}^3$$ Z 3

“…In 2D, a majority of prior work focus on the characterization and generation of circles, rings, discs, and circular arcs [1,2,23,28,29,40,50,51,58,60,65,70]. In 3D, apart from straight lines and planes [14,[16][17][18]25,26,32,37,46,69], several theoretical studies on the characterization of digital spheres and hyperspheres have appeared recently [15,20,21,31,33,36,54,55,66,67].…”

“…There are some related prior work on finding the lattice points on or inside a real sphere of a given radius [15,20,21,31,36,41,55,67], and on finding a real sphere that passes through a given set of lattice points [54]. Most of them are closely related to the determination of lattice points in circles [43], ellipsoids [22,45], or surfaces of revolution [19].…”