| Summary: | 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 $$\begin{equation*} \hspace*{24.5pt}M_{{\bm \alpha}, {\bm \beta}} (T) = \int _T^{2T} \prod _{k = 1}^m {\left|\zeta \left(\tfrac{1}{2} + i (t + \alpha _k)\right)\right|}^{2 \beta _k} dt\hspace*{-24.5pt} \end{equation*}$$ introduced by Chandee Q. J. Math. 62(2011), no. 3, 545–572, where α = α ( T ) = ( α 1 , … , α m ) ${\bm \alpha} = {\bm \alpha} (T) = (\alpha _1, \ldots, \alpha _m)$ and β = ( β 1 … , β m ) $\bm {\beta } = (\beta _1 \ldots, \beta _m)$ satisfy | α k | ⩽ T / 2 $|\alpha _k| \leqslant T/2$ and β k ⩾ 0 $\beta _k\geqslant 0$ . We shall prove M α , β ( T ) ≪ β T ( log T ) β 1 2 + ⋯ + β m 2 ∏ 1 ⩽ j < k ⩽ m | ζ ( 1 + i ( α j − α k ) + 1 / log T ) | 2 β j β k . $$\begin{eqnarray*} \qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\;&&M_{\bm {\alpha },\bm {\beta }}(T) \ll _{\bm {\beta }} T (\log T)^{\beta _1^2 + \cdots + \beta _m^2} \prod _{1\leqslant j < k \leqslant m} |\zeta \big(1 + i(\alpha _j - \alpha _k)\nonumber\\ &&\quad+ 1/ \log T\big)|^{2\beta _j \beta _k}. \end{eqnarray*}$$ This improves upon the previous best known bounds due to Chandee and Ng, Shen, and Wong [Can. J. Math. Published online 2023:1–31. DOI 10.4153/S0008414X23000548], particularly when the differences | α j − α k | $|\alpha _j - \alpha _k|$ are unbounded as T → ∞ $T\rightarrow \infty$ . The key insight is to combine work of Heap, Radziwiłł, and Soundararajan [Q. J. Math. 70 (2019), no. 4, 1387–1396] and work of the author [arXiv preprint arXiv:2301.10634 (2023)] with the work of Harper [arXiv preprint arXiv.1305.4618 (2013)] on the moments of the zeta function.
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