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 Phragmén--Lindelöf principle. In particular, for $\sigma \in [\frac{1}{2},\frac{9}{10}]$, we establish $$|L(\sigma +it,\chi )|\le (1.105)(0.692{)}^{\sigma }{q}^{\frac{31}{80}-\frac{2}{5}\sigma }(\log q{)}^{\frac{33}{16}-\frac{9}{8}\sigma }|\sigma +it|.$$

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