  Let $\chi$ a primitive character$\pmod q$ 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 $s=1/2+it$
