Large value estimates for Dirichlet polynomials, and the density of zeros of Dirichlet's $L$-functions
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918088885141504 |
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| author | Chen, Bin |
| author_facet | Chen, Bin |
| contents | It is proved that \[ \sum_{χ\bmod q}N(σ, T, χ) \lesssim_ε (qT)^{7(1-σ)/3+ε}, \] where $N(σ, T, χ)$ denote the number of zeros $ρ= β+ it$ of $L(s, χ)$ in the rectangle $σ\leq β\leq 1$, $|t| \leq T$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_08296 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Large value estimates for Dirichlet polynomials, and the density of zeros of Dirichlet's $L$-functions Chen, Bin Number Theory Primary 11M06, 11M26, Secondary 11N05, 11N13 It is proved that \[ \sum_{χ\bmod q}N(σ, T, χ) \lesssim_ε (qT)^{7(1-σ)/3+ε}, \] where $N(σ, T, χ)$ denote the number of zeros $ρ= β+ it$ of $L(s, χ)$ in the rectangle $σ\leq β\leq 1$, $|t| \leq T$. 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. |
| title | Large value estimates for Dirichlet polynomials, and the density of zeros of Dirichlet's $L$-functions |
| topic | Number Theory Primary 11M06, 11M26, Secondary 11N05, 11N13 |
| url | https://arxiv.org/abs/2507.08296 |