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

Abstract

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 $\ddot{I}ƒ\le {\hat{I}}^2\le 1$, $|t|\le T$. 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.

58 pages. After the first version by the first author was released, the other authors informed BC that they had independently obtained the exponent $7/3$ in the zero density estimate. The present version is the result of the authors' subsequent collaboration on this topic