ON THE ERROR TERM IN THE GAUSS CIRCLE PROBLEM THEOPHILUS AGAMA Abstract. Using the compression method, we prove an inequality related to the Gauss circle problem. Let Nr denotes the number of integral points in a circle of radius r> 0, then we have 2r 2  1+ 1 4 X 1≤k≤b log r log 2 c 1 2 2k-2  +O( r log r ) ≤Nr ≤ 8r 2  1+ X 1≤k≤b log r log 2 c 1 2 2k-2  +O( r log r ) for all r> 1. This implies that the error function E(r) of the counting function Nr  r 1- for any > 0. 1. Introduction The Gauss circle problem is a classic question in number theory that concerns the approximation of the number of lattice points within a circle in the Euclidean plane. Specifically, the problem asks about the error term in the approximation of the number of lattice points N (r) inside a circle of radius r, where the exact number of points is compared to the area of the circle πr 2 . The main challenge lies in understanding the difference between the exact count of lattice points and the area, known as the error term, and establishing its asymptotic behaviour as r grows large. This problem connects to deep areas of mathematics, such as analytic number theory, geometric analysis, and the distribution of integer solutions to polynomial equations. In this work, we explore the error term in the Gauss circle problem under specific transformations of the lattice points, aiming to refine the asymptotic bounds for the error term and deepen our understanding of the distribution of lattice points in circles. By deriving precise upper and lower bounds, we seek to contribute to the ongoing exploration of the error term in lattice point counting functions. Precisely, the Gauss circle problem is a problem that seeks to counts the number of integral points in a circle centered at the origin and of radius r. It is fairly easy to see that the area of a circle of radius r> 0 gives a fairly good approximation for the number of such integral points in the circle, since on average each unit square in the circle contains at least an integral point. In particular, by denoting N (r) to be the number of integral points in a circle of radius r, then the following elementary estimate is well-known N (r)= πr 2 + |E(r)| where |E(r)| is the error term. The real and the main problem in this area is to obtain a reasonably good estimate for the error term. In fact, it is conjectured that |E(r)| r 1 2 + Date : January 2, 2025. 2010 Mathematics Subject Classification. Primary 11Pxx, 11Bxx, 05-xx; Secondary 11Axx. 1