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

TL;DR

This paper improves bounds on the density of zeros of Dirichlet L-functions using advanced sum estimates, leading to new results on least primes in arithmetic progressions and Goldbach representations.

Contribution

It introduces a sharper zero-density estimate for Dirichlet L-functions by applying the Guth-Maynard method to sums with GCD twists, improving previous exponents.

Findings

Enhanced zero-density bounds with exponent 7/3

New bounds on least primes in prime power moduli

Results on least Goldbach numbers in arithmetic progressions

Abstract

It is proved that \[ \sum_{\chi \bmod q}N(\sigma , T, \chi) \lesssim_{\epsilon} (qT)^{7(1-\sigma)/3+\epsilon}, \] where $N(\sigma, T, \chi)$ denote 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 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.