On high power moments of the error term of the Dirichlet divisor function over primes

Let $3\leqslant k\leqslant9$ be a fixed integer, $p$ be a prime and $d(n)$ denote the Dirichlet divisor function. We use $\Delta(x)$ to denote the error term in the asymptotic formula of the summatory function of $d(n)$. The aim of this paper is to study the $k$-th power moments of $\Delta(p)$, namely $\sum_{p\leqslant x}\Delta^k(p)$, and we give an asymptotic formula.