Optimal Shifting Method in Dirichlet's divisor problem

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.