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

TL;DR

This survey reviews recent advances in the theory of lattice points within large geometric regions and their connections to divisor problems and arithmetic functions, highlighting new methods and recent results in a classical area.

Contribution

It provides a comprehensive overview of recent developments and novel methods in lattice point problems and related arithmetic functions, updating the classical theory.

Findings

Recent progress on circle and sphere problems

New techniques for counting lattice points in complex regions

Advances in divisor problems and their arithmetic implications

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.