The second moment of twisted modular $L$-functions and Dirichlet $L$-functions at the central point

Zhengye Chen, Yongxiao Lin

We prove an asymptotic formula with a power-saving error term for the second moment $$ \sum_{q \in \mathcal{Q}} \; \; \sideset{}{^\flat}\sum_{χ\bmod q} \left| L\big( \tfrac{1}{2} ,…