Lower bounds for shifted moments of the Riemann zeta function

In previous work, the author gave upper bounds for the shifted moments of the zeta function $$M_{\alpha ,\beta }(T)={\int }_T^{2T}\prod _{k=1}^m\mathrm{\mid }\zeta (\frac{1}{2}+i(t+{\alpha }_k)){\mathrm{\mid }}^{2{\beta }_k}dt$$ introduced by Chandee, where $\alpha =\alpha (T)=({\alpha }_1,\dots ,{\alpha }_m)$ and $\beta =({\beta }_1\dots ,{\beta }_m)$ satisfy $\mathrm{\mid }{\alpha }_k\mathrm{\mid }\le T\mathrm{/}2$ and ${\beta }_k\ge 0$. Assuming the Riemann hypothesis, we shall prove the corresponding lower bounds: $$M_{\alpha ,\beta }(T){\gg }_{\beta }T(\log T{)}^{{\beta }_1^2+\cdots +{\beta }_m^2}\prod _{1\le j<k\le m}\mathrm{\mid }\zeta (1+i({\alpha }_j-{\alpha }_k)+1\mathrm{/}\log T){\mathrm{\mid }}^{2{\beta }_j{\beta }_k}\mathrm{.}$$