On the mean square of the Riemann zeta-function in short intervals

It is proved that, for $T^{ϵ}\le G=G(T)\le \frac{1}{2}\sqrt{T}$, $${\int }_T^{2T}(I_1(t+G)-I_1(t){)}^{2}dt=TG\sum _{j=0}^3{a}_j{\log }^j(\frac{\sqrt{T}}{G})+O_{ϵ}(T^{1+ϵ}+T^{1\mathrm{/}2+ϵ}{G}^2)$$ with some explicitly computable constants $a_j(a_3>0)$ where, for a fixed natural number $k$, $$I_k(t,G)=\frac{1}{\sqrt{\pi }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\mathrm{\mid }\zeta (1\mathrm{/}2+it+iu){\mathrm{\mid }}^{2k}{\mathrm{e}}^{-(u\mathrm{/}G{)}^2}du\mathrm{.}$$ The generalizations to the mean square of $I_1(t+U,G)-I_1(t,G)$ over $[T,T+H]$ and the estimation of the mean square of $I_2(t+U,G)-I_2(t,G)$ are also discussed.