Correlations of the Riemann zeta function

Abstract

Assuming the Riemann hypothesis, we investigate 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 $\alpha =\alpha (T)=({\alpha }_1,\dots ,{\alpha }_m)$ and $\beta =({\beta }_1\dots ,{\beta }_m)$ satisfy $|{\alpha }_k|\le T/2$ and ${\beta }_k\ge 0$. We shall prove that \[ M_{α,β}(T) \ll_{β} T (\log T)^{β_1^2 + \cdots + β_m^2} \prod_{1\leq j < k \leq m} |ζ(1 + i(α_j - α_k) + 1/ \log T )|^{2β_j β_k}. \] 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 $T\to \infty$. The key insight is to combine work of Heap, Radziwiłł, and Soundararajan and work of the author with the work of Harper on the moments of the zeta function.

Proof simplified and minor corrections made