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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 anew proof of an upper bound of Heath-Brown for $|L(s,\chi)|$ on the criticalline $s=1/2+it$

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