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 \[ M_{{\alpha},{\beta}}(T) = \int_T^{2T} \prod_{k = 1}^m |\zeta(\tfrac{1}{2} + i (t + \alpha_k))|^{2 \beta_k} dt \] introduced by Chandee, where ${\alpha} = {\alpha}(T) = (\alpha_1, \ldots, \alpha_m)$ and ${\beta} = (\beta_1 \ldots , \beta_m)$ satisfy $|\alpha_k| \leq T/2$ and $\beta_k\geq 0$. Assuming the Riemann hypothesis, we shall prove the corresponding lower bounds: \[ M_{{\alpha},{\beta}}(T) \gg_{{\beta}} T (\log T)^{\beta_1^2 + \cdots + \beta_m^2} \prod_{1\leq j < k \leq m} |\zeta(1 + i(\alpha_j - \alpha_k) + 1/ \log T )|^{2\beta_j \beta_k}. \]