Hybrid moments of the Riemann zeta-function

TL;DR

This paper investigates hybrid moments of the Riemann zeta-function on the critical line, establishing bounds for specific parameter ranges and applying results to Mellin transform mean square estimates.

Contribution

It provides new bounds for hybrid moments of the zeta-function for certain parameter values and explicitly determines the applicable range of G(T).

Findings

Bounded hybrid moments for specific parameters.

Established the range of G(T) where bounds hold.

Applied results to Mellin transform mean square bounds.

Abstract

The "hybrid" moments $$ \int_T^{2T}|\zeta(1/2+it)|^k{(\int_{t-G}^{t+G}|\zeta(1/2+ix)|^\ell dx)}^m dt $$ of the Riemann zeta-function $\zeta(s)$ on the critical line $\Re s = 1/2$ are studied. The expected upper bound for the above expression is $O_\epsilon(T^{1+\epsilon}G^m)$. 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 $|\zeta(1/2+ix)|^4$ is given.