On a Smoothed Walfisz Divisor Problem

This work is in the spirit of our previous investigation on a smooth Dirichlet divisor problem, where we now replace the Dirichlet divisor function $\tau$ by the sum-of-divisors function $\sigma$. We prove a totally explicit asymptotic formula for the sum of $\sigma(n)$ twisted by the weight $1-x/n$, which enables us to eliminate the difficult part in the classical average order of $\sigma(n)$. As a corollary, we deduce the convergence of an integral dealing with the error term in the Walfisz divisor problem. We also provide an appendix containing the necessary explicit results derived from the mean value theorem and the Euler-Maclaurin summation formula.