On the Riemann zeta-function and the divisor problem III

Abstract

Let $Δ(x)$ denote the error term in the Dirichlet divisor problem, and $E(T)$ the error term in the asymptotic formula for the mean square of $|ζ(1/2+it)|$. If $E^*(t) = E(t) - 2πΔ^*(t/2π)$ with $Δ^*(x) = -Δ(x) + 2Δ(2x) - {1\over2}Δ(4x)$ and we set $\int_0^T E^*(t) dt = 3πT/4 + R(T)$, then we obtain $$ R(T) = O_ε(T^{593/912+ε}), \int_0^TR^4(t) dt \ll_εT^{3+ε}, $$ and $$ \int_0^TR^2(t) dt = T^2P_3(\log T) + O_ε(T^{11/6+ε}), $$ where $P_3(y)$ is a cubic polynomial in $y$ with positive leading coefficient.