Lower bounds for shifted moments of the Riemann zeta function

In previous work, the author gave upper bounds for the shifted moments of the zeta function \[ M_{{α},{β}}(T) = ∫_T2T ∏k = 1^m |ζ(1/2 + i (t + α_k))|2 β_k dt \] introduced by Chandee, where α = α(T) = (α₁, …, αₘ) and β = (β₁ … , βₘ) satisfy |αₖ| ≤ T/2 and βₖ≥ 0. Assuming the Riemann hypothesis, we shall prove the corresponding lower bounds: \[ M_{{α},{β}}(T) _{{β}} T (log T)β_1^2 + ⋯ + β_m^2 ∏1≤ j < k ≤ m |ζ(1 + i(α_j - α_k) + 1/ log T )|2β_j β_k. \]