Explicit Subconvexity Estimates for Dirichlet $L$-functions

Abstract: Given a Dirichlet character $\chi$ modulo $q$ and its associated $L$-function, $L(s,\chi)$, we provide an explicit version of Burgess' estimate for $|L(s, \chi)|$. We use partial summation to provide bounds along the vertical lines $\Re{s} = 1 - {r}^{-1}$, where $r$ is a parameter associated with Burgess' character sum estimate. These bounds are then connected across the critical strip using the Phragm\'en--Lindel\"of principle. In particular, for $\sigma \in [\frac{1}{2}, \frac{9}{10}]$, we establish $$|L(\sigma + it, \chi)| \leq (1.105) (0.692)^\sigma q^{\frac{31}{80}-\frac{2}{5}\sigma}(\log{q})^{\frac{33}{16}-\frac{9}{8}\sigma} |\sigma + it|.$$