Non-vanishing of Dirichlet L-functions at the central point

Let $χ$ be a primitive Dirichlet character modulo $q$ and $L(s,χ)$ be the Dirichlet L-function associated to $χ$. Using a new two-piece mollifier we show that $L(\tfrac{1}{2},χ)\ne0$ for at least 34% of the characters in the family.

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