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

It is proved that $$\sum _{\chi q}N(\sigma ,T,\chi ){\lesssim }_{\epsilon }(qT{)}^{7(1-\sigma )\mathrm{/}3+\epsilon },$$ where $N(\sigma ,T,\chi )$ denote the number of zeros $\rho =\beta +it$ of $L(s,\chi )$ in the rectangle $\sigma \le \beta \le 1$, $\mathrm{\mid }t\mathrm{\mid }\le T$. The exponent $7\mathrm{/}3$ improves upon Huxley’s earlier exponent of $12\mathrm{/}5$. 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.