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Let $ \Lambda$ be the von Mangoldt function and $ \chi$ a primitive character mod $ q$ , then we have the explicit formula $ $ \sum_{n \leq X} \Lambda(n) \chi(n) = \delta_{\chi} X – \sum_{ |Im \ \rho| \leq T} \frac{X^{\rho}}{\rho} + O((\frac{X}{T}+1) \log^2(qXT)), $ $ where $ \delta_{\chi}$ is $ 1$ if $Read more