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, {} (T) = T^2T ₊ = ₁ᵐ | (12 + i (t + ₖ) ) |^2 ₖ dt \ introduced by Chandee, where = (T) = (₁, , ₘ) and = (₁, ₘ) satisfy |ₖ| T/2 and ₖ 0. Assuming the Riemann hypothesis, we shall prove the corresponding lower bounds: \ M, {} (T) T (T) ^₁² + + ₘ² ₁ ₉ < ₊ ₌ | (1 + i (ⱼ - ₖ) + 1/ T) |^2ⱼ ₖ. \