On some mean value results for the zeta-function and a divisor problem II

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Let d(n) be the number of divisors of n, let γ denote Euler's constant and Δ(x):= ∑ₙ≤ ₓd(n) - x(x + 2γ-1) denote the error term in the classical Dirichlet divisor problem, and let ζ(s) denote the Riemann zeta-function. It is shown that ∫₀ᵗΔ(t)|ζ(1/2+it)|²dt ≪ T(T)⁴. Further, if 2≤ k≤ 8 is a fixed integer, then we prove the asymptotic formula ∫₁ᵗΔᵏ(t)|ζ(1/2+it)|²dt=c₁(k)T1+ k4 T+ c₂(k)T1+ k4+Oε(T1+ k4-ηₖ+ε), where c₁(k) and c₂(k) are explicit constants, and where η₂= 3/20, η₃= η₄=1/10,η₅=3/80,η₆=35/4742,η₇=17/6312,η₈=8/9433. The results depend on the power moments of Δ(t) and E(T), the classical error term in the asymptotic formula for the mean square of |ζ(1/2+it)|.