On the mean square of the divisor function in short intervals

We provide upper bounds for the mean square integral $${\int }_X^{2X}({\mathrm{\Delta }}_k(x+h)-{\mathrm{\Delta }}_k(x){)}^{2}dx(h=h(X)\gg 1,h=o(x)\text{roman}asX\to \mathrm{\infty })$$ where $h$ lies in a suitable range. For $k\ge 2$ a fixed integer, ${\mathrm{\Delta }}_k(x)$ is the error term in the asymptotic formula for the summatory function of the divisor function $d_k(n)$, generated by ${\zeta }^k(s)$.