Large Value Estimates for Dirichlet Polynomials with Characters and Zero Density of Dirichlet $L$-Functions

Bin Chen, Vishal Gupta, Yung Chi Li

It is proved that \[ \sum_{χ\bmod q}N(σ,T,χ) \ll_ε (qT)^{7(1-σ)/3+ε}, \] where $N(\ddot{I}ƒ,T,\ddot{I}‡)$ denotes the number of zeros $\ddot{I}={\hat{I}}^2+it$ of $L(s,\ddot{I}‡)$ in the rectangle $σ\leq β\le…