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 $N(\sigma ,T,\chi )$ denotes the number of zeros $\rho =\beta +it$ of $L(s,\chi )$ in the rectangle $\sigma \le \beta \le 1$, $|t|\le…