Shifted divisor problem and random divisor problem (Q1312053)

Let \(\alpha\), \(\beta\) be fixed constants with \(0<\alpha\leq 1\), \(0<\beta\leq 1\). The author considers the lattice point problem \(D(x;\alpha,\beta)= \# \{(m,n)\): \(m,n\in\mathbb{N}\cup\{0\}\), \((m+\alpha) (n+\beta)\leq x\}\), where the points with \((m+\alpha) (n+\beta)=x\) are counted with a factor \(1/2\). It is established a Voronoi-type identity and upper and lower bounds for the error term in the asymptotic expansion of \(D(x;\alpha,\beta)\). There are only short outlines of the proofs.