  In 2007, assuming the Riemann Hypothesis (RH), Soundararajan \cite{Moment}
proved that $\int_{0}^T |\zeta(1/2 + it)|^{2k} dt \ll_{k, \epsilon} T(\log
T)^{k^2 + \epsilon}$ for every $k$ positive real number and every $\epsilon >
0.$ In this paper we generalized his methods to find upper bounds for shifted
moments. We also obtained their lower bounds and conjectured asymptotic
formulas based on Random matrix model, which is analogous to Keating and
Snaith's work. These upper and lower bounds suggest that the correlation of
$|\zeta(\h + it + i\alpha_1)|$ and $|\zeta(\h + it + i\alpha_2)|$ transition at
$|\alpha_1 - \alpha_2| \approx \frac{1}{\log T}$. In particular these
distribution appear independent when $|\alpha_1 - \alpha_2|$ is much larger
than $\frac{1}{\log T}.$
