The circle method and shifted convolution sums involving the divisor function

Abstract

Let Q(x) be a positive definite integral quadratic form with the determinant D being squarefree, and r(n,Q) denote the number of representations of n by the quadratic form Q. In this paper, we apply the Hardy-Littlewood-Kloosterman circle method to derive the asymptotic formula for the shifted convolution sum of the divisor function d(n) and Fourier coefficients r(n,Q). With more efforts, our method should have a number of applications for other multiplicative functions.