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

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It is proved that, for Tε≤ G = G(T) ≤ 12√T, ∫ₜ²ᵗ(I₁(t+G)-I₁(t))² dt = TG∑ⱼ₌₀³aⱼʲ (√T G) + Oε(T¹⁺ε+ T¹/²⁺εG²) with some explicitly computable constants aⱼ (a₃>0) where, for a fixed natural number k, Iₖ(t,G) = 1√π∫₋∞∞ |ζ(1/2+it+iu)|²ᵏ e⁻⁽ᵘ/ᵍ⁾² du. The generalizations to the mean square of I₁(t+U,G) - I₁(t,G) over [T, T+H] and the estimation of the mean square of I₂(t+U,G)-I₂(t,G) are also discussed.