Hybrid Moments of the Riemann Zeta-Function

The “hybrid” moments ∫T2Tζ1/2+itk∫t-Gt+Gζ1/2+ixldxmdt Tε≪G=GT≪T of the Riemann zeta-function ζs on the critical line Res=1/2 are studied. The expected upper bound for the above expression is Oε(T1+εGm). This is shown to be true for certain specific values of k,l,m∈N, and the explicitly determined range of G=G(T;k,l,m). The application to a mean square bound for the Mellin transform function of ζ1/2+ix4 is given.