On some mean value results for the zeta-function in short intervals

Abstract

Let $Δ(x)$ denote the error term in the Dirichlet divisor problem, and let $E(T)$ denote the error term in the asymptotic formula for the mean square of $|ζ(1/2+it)|$. If $E^*(t) := E(t) - 2πΔ^*(t/(2π))$ with $Δ^*(x) := -Δ(x) + 2Δ(2x) - \frac{1}{2}Δ(4x)$ and $\int_0^T E^*(t)\,dt = \frac{3}{4}πT + R(T)$, then we obtain a number of results involving the moments of $|ζ(1/2+it)|$ in short intervals, by connecting them to the moments of $E^*(T)$ and $R(T)$ in short intervals. Upper bounds and asymptotic formulas for integrals of the form $$ \int_T^{2T}\left(\int_{t-H}^{t+H}|ζ(1/2+iu)|^2\,du\right)^k\,dt \qquad(k\in N, 1 \ll H \le T) $$ are also treated.