Assuming the Riemann hypothesis, we investigate 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 $\beta_k\geq 0$. We shall prove that \[ M_{ { \alpha},{ \beta}}(T) \ll_{ { \beta}} T (\log T)^{ \beta_1^2 + \cdots + \beta_m^2} \prod_{ 1\leq j < k \leq m} |\zeta(1 + i(\alpha_j - \alpha_k) + 1/ \log T )|^{ 2\beta_j \beta_k}. \] This improves upon the previous best known bounds due to Chandee and Ng, Shen, and Wong, particularly when the differences $|\alpha_j - \alpha_k|$ are unbounded as $T \rightarrow \infty$. The key insight is to combine work of Heap, Radziwi{ \l}{ \l}, and Soundararajan and work of the author with the work of Harper on the moments of the zeta function.
Categories: math.NT