Published online by Cambridge University Press: 01 August 1999
Let E2(T) denote the error term in the asymptotic formula for ∫T0[mid ]ζ(½+it)[mid ]4dt. It is proved that there exist constants A>0, B>1 such that for T[ges ]T0>0 every interval [T, BT] contains points T1, T2 for which
formula here
and that ∫T0[mid ]E2(t) [mid ]adt[Gt ]T1+(a/2) for any fixed a[ges ]1. These results complement earlier results of Motohashi and Ivić that ∫T0E2(t) dt[Lt ]T3/2 and that ∫T0E22(t) dt[Lt ]T2logCT for some C>0. Omega-results analogous to the above ones are obtained also for the error term in the asymptotic formula for the Laplace transform of [mid ]ζ(½+it)[mid ]4.