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

2025/07/11 by Bin Chen, Vishal Gupta, Chen, Bin +1 · 1 citation
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paper · pdf · doi:10.48550/arxiv.2507.08296

Abstract

It is proved that ∑_{χ\bmod q}N(σ,T,χ) ≪_ε (qT)^7(1-σ)/3+ε, where N(σ,T,χ) denotes the number of zeros ρ=β+it of L(s,χ) in the rectangle σ≤ β≤ 1, |t|≤ 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.

Cited by

• Refinements for primes in short arithmetic progressions

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• New large value estimates for Dirichlet polynomials
• Large character sums: Burgess's theorem and zeros of L-functions
• Small zeros of Dirichlet L-functions of quadratic characters of prime modulus
• Large values of character sums with multiplicative coefficients
• Primes in arithmetic progressions to large moduli, and Goldbach beyond the square-root barrier