An asymptotic formula involving the triple divisor function

Abstract

Suppose $d_3\left(n\right)$ denotes the classical triple divisor function. Let $Q(x)$ be a positive definite integral quadratic form, and $r(n,Q)$ denote the number of representations of n by the quadratic form Q. In this paper, we will establish an asymptotic formula of the summation$\sum _{n\le X}{d}_3(n+h)r(n,Q),$ where h is a positive integer satisfying $h\le H\ll X^{1-\epsilon }$. Our result breaks through the trivial bound of the above summation and obtains the power saving in O-term.