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arXiv · 2411.18136

On certain correlations into the divisor problem

Alexandre Dieguez
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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 $\psi^{-1}$.

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Alexandre Dieguez. 2024-11-27. On certain correlations into the divisor problem. https://doi.org/10.1016/j.jnt.2025.08.021

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