On certain correlations into the divisor problem

Abstract

For a fixed irrational $\theta >0$ with a prescribed irrationality measure function, we study the correlation ${\int }_1^X\Delta \left(x\right)\Delta \left(\theta x\right)dx$, where Δ is the Dirichlet error term in the divisor problem. When θ 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 ψ, decorrelation can be quantified in terms of ${\psi }^{-1}$.