Explicit Subconvexity Estimates for Dirichlet $L$-functions

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