On certain correlations into the divisor problem

TL;DR

This paper investigates the correlation of the divisor problem's error term for irrational multiples, establishing how the degree of irrationality influences decorrelation rates, with implications for understanding number theoretic error behaviors.

Contribution

It provides a quantitative analysis of decorrelation rates based on the irrationality measure function of the irrational number involved.

Findings

Decorrelation occurs at rates depending on the irrationality measure.

Strong decorrelation holds for all positive irrationals except possibly Liouville numbers.

Quantifies decorrelation in terms of the inverse of the irrationality measure function.

Abstract

For a fixed irrational $\theta > 0$ with a prescribed irrationality measure function, we study the correlation $\int_1^X \Delta(x) \Delta(\theta x) dx$, where $\Delta$ is the Dirichlet error term in the divisor problem. When $\theta$ has a finite irrationality measure, it is known that decorrelation occurs at a rate expressible in terms of this measure. Strong decorrelation occurs for all positive irrationals, except possibly Liouville numbers. We show that for irrationals with a prescribed irrationality measure function $\psi$, decorrelation can be quantified in terms of $\psi^{-1}$.