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

We obtain, for T ^ε ≤U=U(T)≤T ^1/2−ε , asymptotic formulas for where Δ(x) is the error term in the classical divisor problem, and E(T) is the error term in the mean square formula for . Upper bounds of the form O _ε (T ^1+ε U ²) for the above integrals with biquadrates instead of square are shown to hold for T ^3/8≤U=U(T)≪ T ^1/2. The connection between the moments of E(t+U)−E(t) and is also given. Generalizations to some other number-theoretic error terms are discussed.