Lower bounds for shifted moments of the Riemann zeta function

Abstract

In previous work, the author gave upper bounds for the shifted moments of the zeta function \[ M_{α,β}(T) = \int_T^{2T} \prod_{k = 1}^m |ζ(\tfrac{1}{2} + i (t + α_k))|^{2 β_k} dt \] introduced by Chandee, where $α = α(T) = (α_1, \ldots, α_m)$ and $β = (β_1 \ldots , β_m)$ satisfy $|α_k| \leq T/2$ and $β_k\geq 0$. Assuming the Riemann hypothesis, we shall prove the corresponding lower bounds: \[ M_{α,β}(T) \gg_{β} T (\log T)^{β_1^2 + \cdots + β_m^2} \prod_{1\leq j < k \leq m} |ζ(1 + i(α_j - α_k) + 1/ \log T )|^{2β_j β_k}. \]