Hybrid moments of the Riemann zeta-function

The "hybrid" moments $\int_T^{2T}|ζ(1/2+it)|^k{(\int_{t-G}^{t+G}|ζ(1/2+ix)|^\ell dx)}^m dt$ of the Riemann zeta-function $ζ(s)$ on the critical line $\Re s = 1/2$ are studied. The expected upper bound for the above expression is $O_ε(T^{1+ε}G^m)$. This is shown to be true for certain specific values of the natural numbers $k,\ell,m$, and the explicitly determined range of $G = G(T;k,\ell,m)$. The application to a mean square bound for the Mellin transform function of $|ζ(1/2+ix)|^4$ is given.

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