On certain correlations into the divisor problem

Abstract

For a fixed irrational $\theta >0$ with a prescribed irrationality measure function, we study the correlation ${\int }_1^X\Delta (x)\Delta (\theta x)dx$, 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 ${\psi }^{-1}$.

18 pages

References in corpus (1)

• Convolution of periodic multiplicative functions and the divisor problem