Large value estimates for Dirichlet polynomials, and the density of zeros of Dirichlet's $L$-functions

It is proved that \[ \sum_{χ\bmod q}N(σ, T, χ) \lesssim_ε (qT)^{7(1-σ)/3+ε}, \] where $N(σ, T, χ)$ denote the number of zeros $ρ= β+ it$ of $L(s, χ)$ in the rectangle $σ\leq β\leq 1$, $|t| \leq T$. 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 involving affine transformations with GCD twists, which emerges from our application of the Guth-Maynard method. As corollaries, we obtain two new arithmetic consequences from this zero-density estimate: first, a result concerning the least prime in arithmetic progressions when the modulus is a prime power; second, a result on the least Goldbach number in arithmetic progressions when the modulus is prime.