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arXiv · 2507.08296

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

Bin Chen·Vishal Gupta·Yung Chi Li
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Abstract

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.

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Bin Chen, Vishal Gupta, Yung Chi Li. 2025-07-11. Large Value Estimates for Dirichlet Polynomials with Characters and Zero Density of Dirichlet $L$-Functions. https://arxiv.org/abs/2507.08296

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