Optimal Shifting Method in Dirichlet's divisor problem

TL;DR

This paper introduces an innovative shifting method to improve estimates in Dirichlet's divisor problem by analyzing integer points under a hyperbola, enhancing classical approaches.

Contribution

It develops a new shifting technique for better estimates in Dirichlet's divisor problem, focusing on mean value analysis rather than direct trigonometric sum estimates.

Findings

Achieves optimal estimates in the classical divisor problem.

Introduces a shift-based method for analyzing integer points under a hyperbola.

Enhances understanding of mean value estimates in divisor problems.

Abstract

In this paper, a new method for investigating Dirichlet's divisor problem is developed. For this purpose, integer points under the graph of a hyperbola are studied. Since many investigations in this direction focus on direct estimates of trigonometric sums and are not suitable for studying means, we shall consider shifts with respect to various parameters to define an optimal one. The method allows for obtaining the best possible estimates in the classical divisor problem.