Sharp omega results for the divisor and circle problems

TL;DR

This paper proves sharp omega results for the divisor and circle problems, improving previous bounds and determining the sign of large values using a novel resonance method involving Gamma distribution kernels.

Contribution

It introduces a new resonance method with Gamma distribution kernels to establish sharp omega results and sign determination for the divisor and circle problems.

Findings

Established conjecturally sharp omega results

Determined the sign of large values in the problems

First improvement on Hafner's 1981 omega results

Abstract

We establish omega results for the divisor and circle problems that are conjecturally sharp, while also determining the sign of the large values obtained. This improves on the work of Soundararajan and on the subsequent independent refinements of Sourmelidis and Mahatab, and gives the first improvement on Hafner's 1981 $\Omega_+$ result for the divisor problem and his $\Omega_-$ result for the circle problem. The main new ingredient is a resonance method which works directly with the phase appearing in the Vorono\"i summation formula. This is achieved by replacing the usual positive kernels by a one-sided sectorial kernel, namely the density of a Gamma distribution, whose Fourier transform lies in a suitable sector of the complex plane.