Lattice points in large regions and related arithmetic functions: Recent developments in a very classic topic

Abstract

This is a survey article on the theory of lattice points in large planar domains and bodies of dimensions 3 and higher, with an emphasis on recent developments and new methods, including a lot of results established only during the last few years. It deals with the classic circle and sphere problems, as well as with the present state-of-the-art concerning lattice points in more general regions and bodies. Furthermore, a thorough account is given on divisor problems and related arithmetic functions.

39 pages

References in corpus (1)

• A note on the Laplace transform of the square in the circle problem

Cited by in corpus (22)

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• Another generalization of the gcd-sum function
• A mean-square bound for the lattice discrepancy of bodies of rotation with flat points on the boundary
• The Laplace Transform of the Second Moment in the Gauss Circle Problem
• Lattice points in stretched model domains of finite type in ${\mathbb{R}}^d$
• Mixed $L^p(L^2)$ norms of the lattice point discrepancy
• Explicit Bounds for Entropy Concentration under Linear Constraints
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• Mean square of the error term in the asymmetric many dimensional divisor problem
• The Error Term of the Summatory Euler Phi Function
• Weyl formulae for Schrödinger operators with critically singular potentials
• Variance of lattice point counting in some special shells in ${\mathbb{R}}^d$
• A lower bound for the error term in Weyl's law for certain Heisenberg manifolds, II
• The lattice discrepancy of certain three-dimensional bodies
• Conditional Results for a Class of Arithmetic Functions: a variant of H. L. Montgomery and R. C. Vaughan's method
• On the symmetry of the k-divisors functions in short intervals
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• Counting in generic lattices and higher rank actions