On some mean square estimates for the
zeta-function in short intervals

Abstract. Let \(\Delta(x)\) denote the error term in the Dirichlet divisor problem, and \(E(T)\) the error term in the asymptotic formula for the mean square of \(|\zeta(\frac{1}{2}+it)|\). If \(E^*(t) = E(t) - 2\pi\Delta^*(t/2\pi)\) with \(\Delta^*(x) = -\Delta(x) + 2\Delta(2x) - \frac{1}{2}\Delta(4x)\) and we set \(\int\limits_0^T E^*(t){\,\rm d} t = 3\pi T/4 + R(T)\), then we obtain $$ \int\limits_T^{T+H}(E^*(t))^2{\,\rm d} t \gg HT^{1/3}\log^3T $$ and $$ HT\log^3T \ll \int\limits_T^{T+H}R^2(t){\,\rm d} t \ll HT\log^3T, $$ for \(T^{2/3+\varepsilon}\le H \le T\).