On the divisor function and the Riemann zeta-function in short intervals

Abstract

We obtain, for $T^\epsilon \le U=U(T)\le T^{1/2-\epsilon}$, asymptotic formulas for $$ \int_T^{2T}(E(t+U) - E(t))^2 dt,\quad \int_T^{2T}(\Delta(t+U) - \Delta(t))^2 dt, $$ where $\Delta(x)$ is the error term in the classical divisor problem, and $E(T)$ is the error term in the mean square formula for $|\zeta(1/2+it)|$. Upper bounds of the form $O_\epsilon(T^{1+\epsilon}U^2)$ for the above integrals with biquadrates instead of square are shown to hold for $T^{3/8} \le U =U(T) \ll T^{1/2}$. The connection between the moments of $E(t+U) - E(t)$ and $|\zeta(1/2+it)|$ is also given. Generalizations to some other number-theoretic error terms are discussed.