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{document}$ \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*} $\end{document}

Let x be a large real number, $d\left(n\right)$ be the Dirichlet divisor function and $\Delta \left(x\right)={\sum }_{n⩽x}d\left(n\right)-xlogx-(2\gamma -1)x$. In this paper, we consider a weighted form of the three primes theorem: $S(N;k_1,k_2,k_3):={\sum }_{p_1+p_2+p_3=N}{\Delta }^{k_1}\left(p_1\right){\Delta }^{k_2}\left(p_2\right){\Delta }^{k_3}\left(p_3\right),$ where $p_1,p_2,p_3$ run over primes, $k_1,k_2,k_3\in \{1,2,\cdots ,9\}$, and N is an large odd integer. For the case $k_i\in \{2,3,\cdots ,9\}$$(i=1,2,3)$, the others two $k_j\in \{2,3,4\}$ with $j\ne i$, an asymptotic formula for $S(N;k_1,k_2,k_3)$ has been derived, along with a non-trivial upper bound estimate for $S(N;k_1,k_2,k_3)$ when $min(k_1,k_2,k_3)=1$.

Let $d(n)$ be the Dirichlet divisor function and $Δ(x)$ denote the error term of the sum $\sum_{n\leqslant x}d(n)$ for a large real variable $x$. In this paper we focus on the sum $\sum_{p\leqslant x}Δ^2(p)$, where $p$ runs over primes. We prove that there exists an asymptotic formula.

Suppose $k\geqslant3$ is an integer. Let $τ_k(n)$ be the number of ways $n$ can be written as a product of $k$ fixed factors. For any fixed integer $r\geqslant2$, we have the asymptotic formula \begin{equation*} \sum_{n_1,\cdots,n_r\leqslant x}τ_k(n_1 \cdots n_r)=x^r\sum_{\ell=0}^{r(k-1)}d_{r,k,\ell}(\log x)^{\ell}+O(x^{r-1+α_k+\varepsilon}), \end{equation*} where $d_{r,k,\ell}$ and $0<α_k<1$ are computable constants. In this paper we study the mean square of $Δ_{r,k}(x)$ and give upper bounds for $k\geqslant4$ and an asymptotic formula for the mean square of $Δ_{r,3}(x)$. We also get an upper bound for the third power moment of $Δ_{r,3}(x)$. Moreover, we study the first power moment of $Δ_{r,3}(x)$ and then give a result for the sign changes of it.