A weighted divisor problem and exponential sum

Abstract

In this paper, we investigate a weighted divisor problem involving the exponential sum of $D_{(1)}(n)$, the $n$th coefficient in the Dirichlet series expansion of $\hat{I}{\P }^{\prime }(s{)}^2$. We establish a truncated Voronoï type formula for the error term of ${\sum }_{n\le x}{D}_{(1)}(n)e(nh/k)$, analogous to the results obtained by Jutila. Utilizing this truncated formula, we derive a mean square estimate of the error term. In addition, we study the associated Riesz sum and the corresponding error term, along with its mean square estimate.

33 pages