A weighted divisor problem and exponential sum

TL;DR

This paper studies a weighted divisor problem linked to exponential sums involving derivatives of the Riemann zeta function, deriving formulas and mean square estimates for associated error terms.

Contribution

It introduces a truncated Voronoi-type formula for the error term of a weighted divisor sum involving _{(1)}(n), extending previous results and providing new mean square estimates.

Findings

Derived a truncated Voronoi formula for the error term.

Established mean square estimates for the error term.

Analyzed the Riesz sum and its error term.

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 $\zeta'(s)^2$. We establish a truncated Vorono\"{i} type formula for the error term of $\sum_{n\leq x}D_{(1)}(n)e(nh/k)$, 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.