ON THE ERROR TERM FOR THE FOURTH MOMENT OF THE RIEMANN ZETA-FUNCTION

Abstract

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

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and that ∫^T₀[mid ]E₂(t) [mid ]^adt[Gt ]T^1+(a/2) for any fixed a[ges ]1. These results complement earlier results of Motohashi and Ivić that ∫^T₀E₂(t) dt[Lt ]T^3/2 and that ∫^T₀E²₂(t) dt[Lt ]T²log^CT 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.