Shifted moments of L functions and moments of theta functions

Assuming the Riemann Hypothesis, Soundararajan showed that $\displaystyle{\int_{0}^{T} \vert \zeta(1/2 + it)\vert^{2k} \ll T(\log T)^{k^2 + \epsilon}}$ . His method was used by Chandee to obtain upper bounds for shifted moments of the Riemann Zeta function. Building on these ideas, we obtain, conditionally, upper bounds for shifted moments of Dirichlet $L$- functions which allow us to derive upper bounds for moments of theta functions.