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

It is proved that
\[
\sum_{\chi \bmod q}N(\sigma,T,\chi)
\ll_{\epsilon}
(qT)^{7(1-\sigma)/3+\epsilon},
\] where $N(\sigma,T,\chi)$ denotes the number of zeros $\rho=\beta+it$ of $L(s,\chi)$ in the rectangle $\sigma\leq \beta\leq 1$, $|t|\leq T$. The exponent $7/3$ improves upon Huxley's earlier exponent of $12/5$.
The key innovation lies in deriving a sharp upper bound for sums over affine transformations of functions with a GCD twist, which arises from our adaptation of the Guth--Maynard method. As applications of the zero density estimates obtained in this paper, we derive a new upper bound for the least Goldbach number in arithmetic progressions modulo a prime and establish new results on primes in arithmetic progressions in short intervals, in particular for prime-power moduli.