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

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

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