On mean square of the error term of a multivariable divisor function

Abstract

Let $ \tau(n) $ be the Dirichlet divisor function and $ k\geqslant2 $ be a fixed integer. We give an asymptotic formula of the mean square of$ \begin{equation*} \Delta_k(x) = \sum\limits_{n_1, \cdots, n_k\leqslant x}\tau(n_1 \cdots n_k)-x^kP_k(\log x). \end{equation*} $

Keywords

• divisor function
• mean square
• dirichlet series