On the mean square of the error term for the asymmetric two-dimensional divisor problem. I (Q2655198)

Suppose \(a\) and \(b\) are two fixed positive integers such that \((a, b) = 1\). In this paper, the authors establish an asymptotic formula: \[ \int\limits_{1}^{T}\Delta^{2}_{a,b }(x)dx=c(a,b)T^{\frac{1+a+b}{a+b}}(1+o(1)), \] where \(\Delta_{a,b }(x)\) is the error term of the general two-dimensional divisor problem. This is a conjecture of Ivić. The proof uses the Voronoi type formula and the theorem of Min on exponential sums.