On a Hybrid Fourth Moment Involving the Riemann Zeta-Function

For each integer 1 ≤ j ≤ 6, we provide explicit ranges for σ for which the asymptotic formula ∫ 0 T ζ 1 2 + i t 4 | ζ ( σ + i t ) | 2 j d t ∼ T ∑ k = 0 4 a k , j ( σ ) log k T $$\displaystyle{\int _{0}^{T}\left \vert \zeta \left (\frac{1} {2} + it\right )\right \vert ^{4}\vert \zeta (\sigma +it)\vert ^{2j}dt \sim T\sum _{ k=0}^{4}a_{ k,j}(\sigma )\log ^{k}T}$$ holds as T → ∞, where ζ(s) is the Riemann zeta-function. The obtained ranges improve on an earlier result of the authors. An application to a weighted divisor problem is also given.