On the Gauss Circle Problem

TL;DR

This paper investigates the Gauss circle problem by analyzing Ramanujan's double series of Bessel functions, establishing their uniform convergence, and providing insights into the problem's structure.

Contribution

It introduces a simple lemma to prove uniform convergence of Ramanujan's Bessel series, offering a new approach to the Gauss circle problem.

Findings

Established uniform convergence of Ramanujan's Bessel series

Provided a new perspective on the Gauss circle problem

Simplified the analysis using a straightforward lemma

Abstract

We analyze the double series of Bessel functions given by Ramanujan. Using a very simple lemma we establish the uniform convergence of these series. By this we address to the Gauss circle problem.