On a sum of the error term of the Dirichlet divisor function over primes

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