Correlations of the Riemann zeta function

Assuming the Riemann hypothesis, we investigate the shifted moments of the zeta function M α , β ( T ) = ∫ T 2 T ∏ k = 1 m ζ 1 2 + i ( t + α k ) 2 β k d t $$\begin{equation*} \hspace*{24.5pt}M_{{\bm \alpha}, {\bm \beta}} (T) = \int _T^{2T} \prod _{k = 1}^m {\left|\zeta \left(\tfrac{1}{2} + i (t + \a...