On a bound of Heath-Brown for Dirichlet $L$-functions on the critical line

Abstract

Let $\chi$ a primitive character$(\mathrm{mod}q)$ and consider the Dirichlet $L$-function $$L(s,\chi )=\sum _{n=1}^{\infty }\frac{\chi (n)}{n^s}.$$ We give a new proof of an upper bound of Heath-Brown for $|L(s,\chi )|$ on the critical line $s=1/2+it$