Lower bounds for shifted moments of the Riemann zeta function

TL;DR

This paper establishes lower bounds for shifted moments of the Riemann zeta function, assuming the Riemann hypothesis, extending previous upper bounds and involving correlations between shifted arguments.

Contribution

It proves the first lower bounds for shifted moments of the zeta function under the Riemann hypothesis, complementing prior upper bounds and analyzing the influence of shifts.

Findings

Lower bounds match the order of magnitude predicted by conjectures.

Results depend on the differences between shift parameters.

The bounds involve products of zeta values at shifted points.

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: \[ M_{{\alpha},{\beta}}(T) \gg_{{\beta}} T (\log T)^{\beta_1^2 + \cdots + \beta_m^2} \prod_{1\leq j < k \leq m} |\zeta(1 + i(\alpha_j - \alpha_k) + 1/ \log T )|^{2\beta_j \beta_k}. \]