A mean value result for the fourth moment of \(|\zeta ({1\over 2}+it)|\) II.

Abstract. We prove that, for a fixed \(j\in {\Bbb N}\), there exists \({\sigma}_0 = {\sigma}_0(j)\;(< 1)\) such that $$ \int\limits_0^T\left|{\zeta\left({1\over 2} + it\right)} \right|^4|{\zeta}({\sigma}+it)|^{2j}{\rm d} t \ll_{j,{\varepsilon}} T^{1+{\varepsilon}} $$ holds for \({\sigma}>{\sigma}_0\). We also indicate how to obtain an asymptotic formula for the above integral, for the range of \({\sigma} > {\sigma}_1={\sigma}_1(j)\), where \({\sigma}_0 < {\sigma}_1 < 1\).