A lower bound on high moments of character sums

For any real $k\ge 2$ and large prime $q$, we prove a lower bound on the $2k$-th moment of the Dirichlet character sum \begin{equation*} \frac{1}{φ(q)} \sum_{\substack{χ$\text{ mod }$q\ χ\neq χ_0}} \Big| \sum_{n\leq x} χ(n)\Big|^{2k}, \end{equation*} where $1\le x\le q$, and $\chi$ is summed over the set of non-trivial Dirichlet characters mod $q$. Our bound is known to be optimal up to a constant factor under the Generalised Riemann Hypothesis. We also get a sharp lower bound on moments of theta functions using the same method.