Fractional Moments of Dirichlet $L$-Functions

Let $k$ be a positive real number, and let $M_k(q)$ be the sum of $\mathrm{\mid }L(\frac{1}{2},\chi ){\mathrm{\mid }}^{2k}$ over all non-principal characters to a given modulus $q$. We prove that $M_k(q){\ll }_k\varphi (q)(\log q{)}^{k^2}$ whenever $k$ is the reciprocal $n^{-1}$ of a positive integer $n$. If one assumes the Generalized Riemann Hypothesis then the estimate holds for all positive real $k<2$.