Correlações no problema do divisor de Dirichlet

Abstract

For a fixed irrational θ > 0 with a prescribed irrationality measure function, we study the correlation $\int_1^X \Delta(x)\Delta(\theta x) dx$, 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 $\psi^{-1}$.