On the Riemann zeta-function and the divisor problem IV

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 $Δ^*(x) = -Δ(x) + 2Δ(2x) - {1\over2}Δ(4x)$, then it is proved that $$ \int_0^T|E^*(t)|^3dt \ll_εT^{3/2+ε}, $$ which is (up to `$ε$' best possible) and $ζ(1/2+it) \ll_εt^{ρ/2+ε}$ if $E^*(t) \ll_εt^{ρ+ε}$.